Mathematics, Logic, Set Theory, and Objectivism Options

[PhilE.Stein]

Topic Description: A general discussion of Objectivist ideas of mathematics, especially regarding mathematical logic and set theory.

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I'd like to know more about Objectivist views of mathematics, especially regarding mathematical logic and set theory. I have a number of questions. I encourage anyone with insight to address any of these questions or to offer their thoughts irrespective of my questions or to ask additional questions. I hope for a productive discussion that will probe for strengths and weaknesses of classical mathematical logic and set theory, intuitionist mathematics, Objectivist proposals for mathematics, and other approaches to mathematics.

1. What is the state of the art in Objectivist philosophy of mathematics? What Objectivist writers have made significant progress in this area, and what publications may one view?

2. What synopsis can be given of Objectivist views on mathematics? Has a systematic and rigorous statement of an Objectivist philosophy of mathematics been offered?

3. Are there Objectivist mathematicians who have published mathematics that embody particularly Objectivist mathematical concerns, approaches, or philosophy? Are there proposals for new systems of mathematical logic or for new axioms for set theory or mathematics? Or do Objectivist mathematicians find already existing systems and axioms satisfactory so that Objectivist alternatives are not needed?

4. How would mathematicians implement Objectivist ideas about mathematics? If the ideas are not suggested for implementation in mathematical theories, then in what way is it hoped that the ideas will affect mathematical practice?

5 How do Objectivists view mathematical logic and set theory? Is there a preference for classical mathematical logic, for intuitionist mathematical logic, or for other systems, approaches, or philosophies? Most particularly, if there is an Objectivist critique of classical mathematical logic and set theory, what is the critique and what alternatives do Objectivists offer?

6. Probably the most common definition of 'finite' and 'infinite' among mathematicians these days is:

x is finite if and only if x is equinumerous with a natural number.
x is infinite if and only if x is not finite.
(Note: The subject of the empty set had been raised in another thread. The empty set is finite, according the above definitions, since the empty set is a natural number.)

Also, set theories such as Zermelo-Fraenkel set theory, have an axiom that there exists a set w such that 0 is a member of w, and if n is a member of w, then the union of n and the singleton of n is a member of w.

What objections, if any, do Objectivists have to these formulations?

7. While constructivist and/or intuitionist mathematics might not be a perfect fit for Objectivism, these might be closer to Objectivism than mathematics set in classical mathematical logic. Though I know of Heyting arithmetic, I have not been able to find intuitionist axioms for other concerns, especially the real numbers or for set theory. What are some intuitionist systems for real numbers or for set theory and how do Objectivists regard these proposals?

8. Would non-standard analysis be preferred by Objectivists or does non-standard analysis carry assumptions that are more disagreeable to Objectivism than notions of infinity in standard analysis?

9. I've heard a little about mathematics that allows only finite sets (if I'm putting this correctly). How does such mathematics express the real numbers? How does such mathematics present its own meta-theory? Do such efforts offer alternatives for Objectivists?

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It is not required that in this thread we address all of these questions, but these questions do give us plenty to talk about. And I hope that all participants of this thread are allowed to comment or even digress upon anything that falls in the general area of mathematics or other subjects, such as epistemology or philosophy, that contribute to our understanding of mathematics. As well, I hope participants will be allowed to discuss the particulars of certain matters of mathematics as these may arise in the course of the larger subject.

Thanks in advance for whatever light anyone may shed upon this subject.

[Ed from OC]

One quick note: there is no such thing as an Objectivist theory of Mathematics, or of Gardening, Knitting, Evolution, Television, etc. There are Objectivists who apply their knowledge of Objectivism to the fields of interest to them. The theories they come up with are their own work, and whether or not those new theories are consistent with Objectivism (A) doesn't affect the truth or falsehood of Objectivism and ( can be a matter of some contention among Objectivists. Just as Objectivists agree on the philosophic principles that apply when voting for President but disagree when it comes to the best application of those principles during an election, Objectivists can disagree about particular scientific or mathematical theories that claim to be consistent with Objectivism.

I don't regard this as a minor issue, and want to be sure a potentially interesting thread doesn't start off on a false premise.

[PhilE.Stein]

Thank you, Ed from OC. You point is well taken and is an important qualification. I did not mean to imply that there is a single Objectivist view on this subject. And I hope to hear of more than one. However, I have heard that there are Objectivists (Pat Corvini was mentioned in another thread, though I could be mistaken in my impression that her views on the subject are related to Objectivism) who apply Objectivist philosophy to the subjects of mathematics and the philosophy of mathematics. I'm interested in knowing more about that, as well as what other Objectivists think about this subject in terms of their own application of Objectivism. In that sense, it is difficult to talk about this without using expressions such as "an [notice, not 'the'] Objectivist view on mathematics", though, again, such expressions should not be understood to mean that there can be only one Objectivist view or even that an individual Objectivist cannot comment upon mathematics outside the context of Objectivism.

[Stephen Speicher]

While constructivist and/or intuitionist mathematics might not be a perfect fit for Objectivism, these might be closer to Objectivism than mathematics set in classical mathematical logic.

I'm curious. Why would you think that?

Though I know of Heyting arithmetic, I have not been able to find intuitionist axioms for other concerns, especially the real numbers or for set theory.

Look up L.E.J Brouwer. Heyting edited Brouwer's Collected Works. See especially Brouwer's set theories of 1918 and 1920.

[PhilE.Stein]

Thanks, Stephen Speicher for your reply.

Just to be clear, as I posted, I don't hold that constructivism and/or intuitionism are, but only that they might be, closer to Objectivism than is classical mathematics. In certain respects, as I'll mention, they might be further apart.

Intuitionists oppose the law of excluded middle applied to infinite sets. That restriction seems to be friendlier to Objectivism in putting a constraint on the classicist's universally generalizing (combined with exclusive middle) about infinite sets. And the intuitionist's rejection of the law of excluded middle might not conflict with Objectivist commitment to the law, since an Objectivist might view the law as not applicable to certain kinds of statements that might be considered in the same terms as what Objectivists call 'the arbitrary'. In other words, certain universal generalizations about infinite sets might be considered arbitrary and hence outside the scope of the law of excluded middle, and hence there would be some common ground between Objectivism and intuitionism.

On the other hand, it seems to me that certain intuitionist semantics might be unacceptable to Objectivism on account of even more elaborate possible world methods than are needed for classical semantics. This strikes me since, as I understand, the notion of possible worlds is not something Objectivists favor, at least not at a surface view. And that raises another question: What would an Objectivist semantics be, since even classical model theory is, at least roughly speaking, based on "possible worlds."

Even more fundamentally, for Objectivists such as Peikoff who reject the necessary/contingent distinction, what is their understanding of logical entailment itself (as found in mathematical logic)? How is the concept of logical entailment even expressed if not through such concepts as necessary and contingent? (I confess not understanding Quine's attack on the distinction, so this question applies not just to Objectivism, though, at least Quine provides relief with a notion of form, which, if I recall correctly, is rejected by Peikoff as he discusses virtually the same notion).

But back to infinity: Since constructivism and intuitionism have a hostility to "completed" infinity, as does Objectivism, and since constructivism, intuitionism, and Objectivism all prefer "potential" infinity, there is a salient alignment on that axis.

By the way, this raises for me the question of how one would formalize a distinction between actual and potential. I've read some of Heyting's thoughts written more than thirty five years after the works by Brouwer you mentioned. I did not find what I was looking for in those particular remarks by Heyting, so I hope I'll be able to find the Brouwer collection in a library. I didn't know that he had proposed axioms for the reals or a set theory, let alone as early as 1918, so I am well served by your reference.

I emphasize that I do not mean to speak for Objectivism or intuitionism and that if my understanding of them in this context is mistaken, then I welcome being informed. I have been able to find only a few remarks on this subject in the main Objectivist texts, so I am admittedly speculating about some of this. I hope that posters will either verify that my speculations are correct or tell my why they are not. Also, I'm speaking rather informally here, so if any technical corrections need to be made, I welcome them too.

[Capitalism Forev...]

x is finite if and only if x is equinumerous with a natural number
[...]
the empty set is a natural number[right][snapback]13363[/snapback][/right]

I think most Objectivists would reject the notion that "each natural number IS a set." A proper mathematics defines natural numbers by reference to what they represent in reality--counts of items--rather than arbitrarily declare some sets to "be" natural numbers.

And while I'm at it: Sets themselves should also be defined by reference to reality. I do NOT think that "set" is a primary concept. It may be one of the starting points of mathematics, so you could consider it a primary with respect to the field of mathematics, but all that means is that "in my hierarchy of knowledge, my definition of the concept 'set' does not fall into the area I call 'mathematics.'"

A set is a mental grouping of objects, arising either as an enumeration or as a combination of a base class and a predicate. For example, when I say, "I'll need to change my oil, my spark plugs, and my air filter," I am establishing a set by enumeration. When I say, "I like red apples," I am using a base class (apples) and a predicate (red) to name a set. Neither example is very mathematical, is it? But both are examples of what mathematics should properly study under the heading "set theory."

[Stephen Speicher]

By the way, this raises for me the question of how one would formalize a distinction between actual and potential. I've read some of Heyting's thoughts written more than thirty five years after the works by Brouwer you mentioned. I did not find what I was looking for in those particular remarks by Heyting, so I hope I'll be able to find the Brouwer collection in a library. I didn't know that he had proposed axioms for the reals or a set theory, let alone as early as 1918, so I am well served by your reference.

You should not be surprised by the date; the roots of Brouwer's intuitionism goes back to his even earlier 1907 doctoral thesis On the Foundations of Mathematics. And, much earlier than that, the first real intuitionist was none other than the great Leopold Kronecker, where you will find him hawking his wares as early as the 1870s. Note, though, Brouwer later sought to distance himself from the earlier intuitionists. In an incomplete draft of an analysis of Abraham Fraenkel's 1927 book Zehn Vorlesungen ueber die Grundlegung der Mengenlehre, Brouwer writes:

"The old-intuitionist assumes intuitively only the natural numbers, but arrives, however, at analysis and geometry by means of classical logic, usually under the application of axioms (e.g. for projective geometry: any two lines either coincide, or have one point in common). Although they usually let a vague intuition suggest these axioms, this intuition then should on the basis of beliefs in a priority give dispensation from the consistency proof, and it has also completely paralyzed the alertness for the absence of constructivity in the rules.

"Therefore the quotation of Kronecker can very well be applied to old-intuitionists. But in the end they are much closer to the formalists than to neo-intuitionism, in particular since the formalists, too, presuppose the natural numbers. These three categories are better characterized as Leibnizians, Kantians and Alogicism. Compared to the 'alogical' or 'correct' mathematics the 'logical' (Kantian or Leibnizian) mathematics appears absurd, chaotic, or as a vague. respectively distorted, image of a (usually also finer and more complicated) reality. The recording of reality by means of axioms and logic has in mathematics exactly as ... only a coarsening and smothering effect."

Brouwer goes on to berate Fraenkel for mixing up these various groups, but his disassociation from the earlier intuitionists is clear. And, relevant to your questions (which I chose not to address) about Brouwer and an Objectivist approach, though I do not presume to speak for Ayn Rand, who, as far as I know never commented on Brouwer, I am one Objectivist who would distance myself from his work even further than Brouwer chose to distance himself from the earlier intuitionists.

[PhilE.Stein]

Thanks, Free Capitalism, for your remarks.

A proper mathematics defines natural numbers by reference to what they represent in reality--counts of items--rather than arbitrarily declare some sets to "be" natural numbers.[right][snapback]13378[/snapback][/right]

Yes, the arbitrariness of the set theoretic definitions of not just 'natural number' but other definitions as well, such as that of 'ordered pair', is something of a thorn even for some set theorists.

As we know, years ago, less arbitrary definitions were hoped for. In particular, for example, '2' would be defined as the set of all sets that have two members (this is not circular, since 'having two members' is previously defined in terms of correspondence). However, as we know, such a definition, though less arbitrary than, for example, the Zermelo or von Neumann definitions, allows set theoretic paradoxes.

We need to keep in mind that set theorists seek definitions for application to formal axiomatic systems that start only with axioms for primitive symbols. So it is quite a tall order that the definitions also "reference what [their definienda] represent in reality". We may not be able to capture in formal languages all that may be bundled in the demand for referencing representations of reality. Perhaps we might think of such arbitrary definitions as we think of arbitrary units of measurements. One might pick up a stick and say, arbitrarily, "Okay, we'll measure off our little camping area with the length of this stick as our unit." So, a mathematician might say, "Okay, I need a way to define each natural number. Here's this device - taking the union of a set and its singleton - which is arbitrary, but it works."

On the other hand, one might reject that mathematics should be performed in formal languages. But formality is what ensures that mathematics is not just a kind of very convincing sounding prose. The requirement of effective method - say upon Church's thesis - ensures that mathematics is indeed objective ('objective' here is not meant to imply 'Objectivist'). One may reject a set of axioms, but one cannot correctly dispute that a proof (an actual proof, not just a purported proof) from those axioms is a proof. Then one is free to choose those theories - even if not in classical mathematical logic - that one thinks best represent reality or are even interesting onto themselves as abstract entities.

Moreover, it doesn't seem that the formal theories interfere with any physical tasks we need to accomplish. Set theory provides a formal theory within which to express real analysis and other mathematics, which includes all kinds of infinities - sets, sequences, et. al. And with this mathematics we make jet planes fly. One may argue that analysis could do without set theory. But some mathematicians do find that their conception is made clearer by foundations. Just about any book on analysis, abstract algebra, topology, probability, statistics, or game theory, begins with a section on set theory. This theory, which includes arbitrary definitions, infinities, et. al, has a part in making jet planes fly, microwave ovens bake, and computers compute. So this theory might not be doing such a bad job at referencing reality, even if in the theory's own odd way.

By the way, I don't mean to insist on the philosophy of formalism, though I am impressed with Curry's argument that formalism allows philosophical neutrality. And I think formalism and Hilbert have been misunderstood and misrepresented. For example, the famous quote found all over the Internet about "game of meaningless symbols" does not have a citation I am aware of. Martin Davis has requested a citation in his Foundations of Mathematics online forum, but none has been produced. Hilbert clearly was very concerned with meaning and mathematical mentation. What formalism offers is not that mathematics is nothing but symbol manipulation, but rather that mathematics should be, in one regard, strict symbol manipulation, without demanding that this is the only regard in which mathematics is understood. We do think about what the symbols mean, but we don't allow that meaning to afford exceptions to the rules we've set up for the manipulations of the symbols.

[...] in my hierarchy of knowledge, my definition of the concept 'set' does not fall into the area I call 'mathematics.'[right][snapback]13378[/snapback][/right]

Fair enough. But I think many (most?) set theorists do not intend for their own use of the terms to supersede other uses, just as you mentioned that sets in the mathematical sense are but a primary with respect to mathematics only. Though, realist mathematicians who believe there are sets and truths about these sets, independent of human thought, might be a fair target for your remark. And I think Curry does offer a reasonable way of avoiding commitment to such realism.

Also, there are theories that do have 'set' as a predicate, but set theories such as ZF, which is the most common, don't even mention sets within the theory itself. Mathematicians do say things like "If we have set, then we have the set of all its subsets." But, if we like, we could say instead, "For all x, there exists a y, such that for all z, z is an element of y if and only if, for all t, if t is an element of z then t is an element of x." And this can be carried even into model theory - a part of the meta-theory. Talking about sets makes it easier to visualize the relations in the theory, but is not required.

A set is a mental grouping of objects, arising either as an enumeration or as a combination of a base class and a predicate.[right][snapback]13378[/snapback][/right]

Then we'll need definitions of 'grouping', 'class' and 'predicate'.

For example, when I say, "I'll need to change my oil, my spark plugs, and my air filter," I am establishing a set by enumeration. When I say, "I like red apples," I am using a base class (apples) and a predicate (red) to name a set. Neither example is very mathematical, is it? But both are examples of what mathematics should properly study under the heading "set theory.[right][snapback]13378[/snapback][/right]

I don't find those examples to be so unmathematical. They seem similar enough to the kind of thing set theory does. ZF allows making sets as you describe as by enumeration. And, informally speaking, the central axiom of Z set theory is that once you have a set and a predicate expressible in the formal language, then you have a subset of that set carved out by the restriction of the predicate. Per your example, If you have the set of apples, and you have defined 'red' in the language, then you have the set of red apples.

I do wonder if Objectivism and ZF don't see eye to eye on two points: (1) In ZF, any finite collection you can gather is just fine as a set, no matter how arbitrary the gathering, and (2) In ZF, predicates don't need to be what Objectivism calls 'essential' or even subject to any rational scrutiny other than that they are expressible as propositional functions in the language.

And another big question is what about infinite sets? Set theory postulates that at least one infinite set exists, from which we derive (in combination with the other more basic, and less controversial axioms) that (1) the set of natural numbers exists and (2) a whole bunch of other wacky infinite sets exist.

And we need some of those infinite sets to express mathematics such as that of the real numbers. But if we are to find some workaround so that we don't refer to these infinite sets, then just what are these workarounds, how are they formalized, and can we avoid making the formalizations more complicated than they're worth? In other words, set theory - with its infinite sets - works, and it's pretty simple (is there anything simpler that does not have infinite sets but gets the job done?), so what is the substantive objection really to talking about these x's and y's, et. al, especially since we don't even have to use the word 'infinity' but rather can confine ourselves just to the variables and the one primitive predicate (member of)?

I think that eschewing arbitrariness and distinguishing essential properties makes sense. But it seems to me that Objectivism may have its work cut out for it in order to have a method of distinguishing essential from non-essential, let alone making that distinction and other requirements applicable to mathematics. Whatever progress has been made is of interest to me.

[PhilO]

Some time ago I had an insight which I think is interesting, regarding the connection of mathematics to reality. (I think it's interesting anyway, but I had zero feedback from the Objectivist mailing list I mentioned it on at the time, so I assume everyone thought it was a big yawn. Be that as may...)

I was thinking of differences between types of numbers: integers, and non-integers (most generally, the real numbers, keeping this out of the realm of complex numbers for now), and wondering what the difference in reality is between them. As I recall I was partially motivated by thinking about the infamous Fermat's last theorem with its focus on the (impossibility for n>2) integer solutions to his equation.

I concluded that epistemologically, there is in fact an interesting distinction, and mapping, that can be made between the two types of numbers. Integers represent, in a sense, the counting of discrete entities. There is really no such thing as a "half entity", or any other non-integer number of entities: there are only instances of particular entities, which may differ in *specific* measurements but either are, or are not, entity-instances. And the number of entities is also a physical invariant, independent of various perspectives that may arise from differences in velocity, etc., which is another interesting fact.

This is as distinct from the real numbers. I concluded that these really represent measurement of attributes, which may and usually do exist as a non-integral relationship to a reference unit of measurement for a given attribute.

So to summarize: integers -> measurement of entity count (i.e. entity existence), reals -> measurement of entity attributes.

For what it's worth.

[PhilE.Stein]

Stephen Speicher, that's some interesting history. However, I knew that Brouwer goes back that far, but I had commented that I didn't know he had given axioms. And you quote him as disdaining axioms.

Why do you have antipathy for intuitionism? You mention that Brouwer distanced himself from earlier constructionists. Are you favorable to the outlook of the early constructionists, or do you distance yourself from their philosophy as well as Brouwer's?

Also, isn't there a similarity between "a finer and more complicated reality" and an Objectivist concern (if it is an Objectivist concern) that mathematics render reality and not depart from reality through formal derivations from abstract and arbitrary axioms?

By the way, emending a comment I made about semantics for intuitionism, another angle occurred to me when reading Dirk van Dalen's Logic And Structure. He gives an informal motivation for Kripke semantics as reflecting a mathematician's "extending both his knowledge and his universe of objects over the course of time." This seems similar to the Objectivist notion of ever growing contextual knowledge. This may be a superficial similarity, though, and I am not claiming that Objectivism and intuitionism are compatible. Whether they are, I just don't know. I'd like to hear more about why one would think that they are not.

[PhilE.Stein]

OliverComputing, your interesting ruminations are not far from the observation that the set of real numbers is equinumerous with the power set of the set of natural numbers. If each subset of the naturals were thought of as a property (though not each one is expressible in a countable language), then we woud have a 1-1 correspondence between the set of reals and the "set of properties of naturals".

[Burgess Laughlin]

Perhaps we might think of such arbitrary definitions as we think of arbitrary units of measurements. One might pick up a stick and say, arbitrarily, "Okay, we'll measure off our little camping area with the length of this stick as our unit." So, a mathematician might say, "Okay, I need a way to define each natural number. Here's this device - taking the union of a set and its singleton - which is arbitrary, but it works."
[right][snapback]13401[/snapback][/right]

I am not a mathematician, so I cannot participate in this discussion. However, I am very interested in the idea of "arbitrary." I am puzzled by why you think that using a particular stick as a unit of measurement is arbitrary. Would you explain? Perhaps a definition of "arbitrary" would help -- in its philosophical meaning and, if there is a special one, in its mathematical meaning.

[Stephen Speicher]

By the way, emending a comment I made about semantics for intuitionism, another angle occurred to me when reading Dirk van Dalen's Logic And Structure. He gives an informal motivation for Kripke semantics as reflecting a mathematician's "extending both his knowledge and his universe of objects over the course of time." This seems similar to the Objectivist notion of ever growing contextual knowledge. This may be a superficial similarity, though, and I am not claiming that Objectivism and intuitionism are compatible. Whether they are, I just don't know. I'd like to hear more about why one would think that they are not.
[right][snapback]13408[/snapback][/right]

I've read a decent amount of van Dalen's work and actually have a lot of respect for his historical analyses. Kripke, wrong as he is, is probably the current best of his lot. And, yes, the connection you identify is a superficial similarity.

I've had some extended conversations about this with a couple of rather famous logicians, and I learned that unless they become as familiar with the principles and details of Objectivism as I have become with the history and technical aspects of their field, that all the time and effort I invested was a waste. I am friendly with a few mathematical logicians whom I respect for the knowledge they have and the work that they do, but they are so steeped in the prevailing view that they cannot punch their way out of the philosophical box that surrounds them. If you really want answers to some of the questions you raise, I suggest you first perform a systematic and concentrated study of the philosophy of Objectivism, starting with Ayn Rand's Introduction to Objectivist Epistemology. Come back when you have mastered that work, and at least the first five chapters of Peikoff's Objectivism: The Philosophy of Ayn Rand, and then perhaps we can meaningfully discuss the incompatibility of intuitionism and Objectivism and the like.

[PhilE.Stein]

NOTE: In an earlier post I mistakenly responded as if to a poster nicknamed 'Free Capitalism' when the actual poster's nickname is 'Capitalism Forever'.

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I am very interested in the idea of "arbitrary." I am puzzled by why you think that using a particular stick as a unit of measurement is arbitrary. Would you explain? Perhaps a definition of "arbitrary" would help -- in its philosophical meaning and, if there is a special one, in its mathematical meaning.
[right][snapback]13411[/snapback][/right]

Thanks for the question, Burgess Laughlin. Capitalism Forever raised the problem of arbitrariness in mathematical definitions. I recognize, without being philosophically precise, that there is an arbitrariness in the von Neumann definition of the natural numbers. I am not too bothered by that arbitrariness, but I am sympathetic to a wish to avoid arbitrariness if possible. So as to a philosophical definition of 'arbitrary', I should think that that task redounds to whomever introduces the term, such as in this thread.

In the everyday sense of 'arbitrary', in the example I gave, the act of making a measurement was not an arbitrary act, but the choice of the particular stick was arbitrary. Many other sticks, of different length even, would work as well as the particular stick chosen. So what is important in this instance is not the particular stick, or the particular unit of measurement, but rather that some stick, or some unit of measurement, was chosen (and that we stick with that stick as our unit). In set theory, say we need a definition, as we might need a unit of measurement at a campsite. But we don't seem to have a "golden measuring rod", "a golden book of perfect definitions"; We don't seem to have a non-arbitrary way of making our definition. So we adopt a definition that, though it is arbitrary, it gets the job done, as the arbitrarily chosen stick gets the job done. I do not mean to suggest this as an argument by analogy to support the arbitrariness of definitions in mathematics but rather to suggest at least some plausibility or a reasonable apology for an admittedly imperfect, yet powerful and efficient conceptual device.

As to a more precisely philosophical sense of 'arbitrary', I am interested in Objectivist ideas. However, these seem to depend on distinguishing between essential and non-essential properties. I am familiar with the explanation (if I am stating it correctly) that an (or 'the') essential property of an entity is that property upon which the greater number of other properties of the entity depend. But what is meant by a property depending (or whatever precise term is used) on another? How does one calculate these dependencies? How many properties are there of an existent? Even if there are only finite number of properties of an entity, how would one peer through the combinatorial possibilities to see which is the one most depended upon by the others? If the notion of necessity is brought in, then does that not entail that some properties are not necessary but only contingent? And I suspect that such talk would not be able to avoid recognizing necessary and contingent truth. And, more fundamentally, what is an Objectivist notion of logical consequence that can serve as a foundation for mathematical reasoning?

As to a mathematical definition of 'arbitrary', I don't have one. Mathematical logic allows instantiation to certain variables, so that one says things like, "Let x be an arbitrary natural number." But that's just lingo for application of the rule of universal instantiation. Also, the notion of effective method demands that each step in an algorithm be a "determined" (there's a better word, but it's not coming to me now); An algorithm may not allow one to have to make a choice even. So one cannot be presented with having to, say, arbitrarily move from square one to square three. Then, of course, there's the big daddy of all arbitrariness: the axiom of choice. But even with the axiom of choice, there is no need, within the theory, to refer to 'arbitrary'.

[Burgess Laughlin]

In the everyday sense of 'arbitrary', in the example I gave, the act of making a measurement was not an arbitrary act, but the choice of the particular stick was arbitrary. Many other sticks, of different length even, would work as well as the particular stick chosen. [...]

As to a more precisely philosophical sense of 'arbitrary', I am interested in Objectivist ideas. [...]

As to a mathematical definition of 'arbitrary', I don't have one. [...]
[right][snapback]13419[/snapback][/right]

[Bold added for emphasis.]

What is "the everyday sense"? If I have understood you correctly, you don't have a definition of "arbitrary" -- either everyday, philosophical, or mathematical -- but you continue to use the term "arbitrary" to label an unknown idea applying to mathematics.

So as to a philosophical definition of 'arbitrary', I should think that that task redounds to whomever introduces the term, such as in this thread.

Why only to the initiator? Why wouldn't the responsibility of definition -- at whatever level -- fall to anyone using the term as a label for an idea? Shouldn't the user always be able to point to reality and say, "This is what I mean by the term/idea"?

The issue isn't merely a dialectical one -- that is, who is responsible for definition in the circumstance of a discussion -- but a basic cognitive one. How can one think about a subject without knowing what the terms/ideas refer to in reality?

[Capitalism Forev...]

As we know, years ago, less arbitrary definitions were hoped for. In particular, for example, '2' would be defined as the set of all sets that have two members[right][snapback]13401[/snapback][/right]

But that's still groping in the wrong direction. To define "2," you need to ask: What fact of reality do we name when we say "2" ? For example, if I say "I have two apples," in what circumstances is that true and in what circumstances is that false?

If I had an apple but that was all, then my statement would be false. If I had an apple and then I had another apple, my statement would be true. So "two apples" means "an apple, and then another apple."

Now, apples are not the only things that can be counted. You can count oranges, sticks, men, sunrises, cows, sheep ... you name it. The meaning of "two oranges," "two sticks," etc. is very similar to the meaning of "two apples" ; this similarity allows us to form an abstraction where we omit the name of the class of items we are counting and only specify the number. So, from "two apples," "two cows," "two hands," "two stars," and so on, we abstract "two." The kind of items we count is different in each case, but the count--"an item, and then another item"--is the same.

"Two" can therefore be defined as follows: "A count where, after identifying one item, I can identify another item."

We need to keep in mind that set theorists seek definitions for application to formal axiomatic systems that start only with axioms for primitive symbols.[right][snapback]13401[/snapback][/right]

My definition of "two" above can be formalized as follows:

CODE

|A| = 2 <=>
   exists x, y such that x element-of A and y element-of A and x != y
   and not exists z such that z element-of A and z != x and z != y

where A is any set.

We may not be able to capture in formal languages all that may be bundled in the demand for referencing representations of reality.[right][snapback]13401[/snapback][/right]

Once you understand the Objectivist view of abstractions, you'll see that a formal system need not capture everything about reality in order to describe reality correctly. All it needs to capture is those elements that are included in the abstractions; it need not concern itself with the facts that are omitted from the abstractions. You need not know anything about the details of apples, cows, etc. in order to make correct statements about the number 2, as those details are what we omit when forming the abstraction "two." What you do need to know is the part we include in the abstraction: the part about there being an item, and there being another item. And that part can be formalized, as I did above.

BTW, note that I have nothing against establishing a correspondence--a bijection--between natural numbers and sets formed by the above method, or by whatever other method. What I object to is saying that the number 2 IS the set {{},{{}}}.

2 = {{},{{}}} is wrong
f(2) = {{},{{}}} is fine

Perhaps we might think of such arbitrary definitions as we think of arbitrary units of measurements. One might pick up a stick and say, arbitrarily, "Okay, we'll measure off our little camping area with the length of this stick as our unit." So, a mathematician might say, "Okay, I need a way to define each natural number. Here's this device - taking the union of a set and its singleton - which is arbitrary, but it works."[right][snapback]13401[/snapback][/right]

When you measure the length of something with a stick, you are establishing a relationship between two facts of reality: the length of the thing you measure and the length of the stick. When you define a number to be a set, you are declaring that the meaning of a symbol should be something you just invented. In the first case, your symbol--the length unit "stick"--references something in reality. In the second case, your symbol "2" references something in your consciousness, even though the practical use of that symbol is to count things in reality.

Objectivism holds that there should be no split between practice and theory. A theory that consists of manipulating symbols that are not in any way attached to reality is not a theory worth pursuing.

Continued in next post...

[Capitalism Forev...]

Moreover, it doesn't seem that the formal theories interfere with any physical tasks we need to accomplish.[right][snapback]13401[/snapback][/right]

That is because people still recognize the correct meaning of mathematical concepts. Imagine if someone took 2 = {{},{{}}} literally:

"The plane will have 2 engines."
"What do you mean 2 engines? 2 is {{},{{}}}. I have a degree in engineering, but I have never heard of a set-that-contains-the-empty-set-and-a-singleton-containing-the-empty-set engine."

The reason the engineers understand the phrase "2 engines" is that they know 2 means "an item, and then another item."

One may argue that analysis could do without set theory. But some mathematicians do find that their conception is made clearer by foundations.[right][snapback]13401[/snapback][/right]

I have nothing against foundations, as long as they are the right ones!

Fair enough. But I think many (most?) set theorists do not intend for their own use of the terms to supersede other uses, just as you mentioned that sets in the mathematical sense are but a primary with respect to mathematics only.[right][snapback]13401[/snapback][/right]

Perhaps I should clarify what I meant by "primary with respect to mathematics only." I did not mean that the word should have two meanings, one used outside mathematics and another used in mathematics. What I meant is that the word should be defined outside mathematics (with the definition possibly repeated in the introductory section of mathematics, before the formal section) and in the formal section it should be treated as a given, without offering a formal definition. The meaning of the word within the formal section should be the same as outside it.

So, when I formally define the union of two sets as

CODE

for all x, x element-of A U B <=> x element-of A or x element-of B

where A, B are sets

then the meaning of "where A, B are sets" is "where A and B refer to mental groupings of objects, arising either as enumerations or as combinations of a base class and a predicate."

And the meaning of the definition is that the union of two such mental groupings is a third mental grouping that includes those and only those objects that are included in either original grouping.

Then we'll need definitions of 'grouping', 'class' and 'predicate'.[right][snapback]13401[/snapback][/right]

Grouping is the action whereby one selects a number of things for performing the same action on each.

A class is a cognitive device used in identifying objects, specified by a concept and possibly some further qualifiers, and capturing an aspect of the nature of the objects in question.

A predicate is a construct in a language that specifies attributes of or actions performed by an object to be named.

I don't find those examples to be so unmathematical. They seem similar enough to the kind of thing set theory does.[right][snapback]13401[/snapback][/right]

My point was that they were practical statements, not theoretical ones. The fact that set theory does something similar is what makes set theory useful in practice.

I do wonder if Objectivism and ZF don't see eye to eye on two points: (1) In ZF, any finite collection you can gather is just fine as a set, no matter how arbitrary the gathering[right][snapback]13401[/snapback][/right]

As a philosophy, Objectivism itself does not treat set theory problems, so the question is rather whether the above approach could be taken by a set theory built on proper Objectivist principles. I think it definitely could.

(2) In ZF, predicates don't need to be what Objectivism calls 'essential' or even subject to any rational scrutiny other than that they are expressible as propositional functions in the language.[right][snapback]13401[/snapback][/right]

Essentiality is a criterion for what to include in a definition of a concept. For specifying a set, there are other criteria: The predicate should be applicable to the base class (you can use the predicate "red" with apples, but you cannot use it with air molecules, since air molecules do not have a color) and decidable for every instance of the base class.

The latter point is important. It makes the infamous "all sets that do not include themselves" invalid as a set specification, because the predicate is undecidable for the set being defined due to circularity.

And another big question is what about infinite sets? Set theory postulates that at least one infinite set exists, from which we derive (in combination with the other more basic, and less controversial axioms) that (1) the set of natural numbers exists and (2) a whole bunch of other wacky infinite sets exist.[right][snapback]13401[/snapback][/right]

An Objectivist mathematician does not need any such derivation to know that the set of natural numbers exists. He knows that counts exist in reality; he knows that they are natural numbers; he knows that he can mentally group natural numbers and that a mental grouping specified by a base class (and, in this case, a vacuously true predicate) is a set--therefore, he knows that the set of natural numbers exists.

[PhilE.Stein]

I don't take responsibility for a definition of 'arbitrary', since there's no position I've taken here that depends on the term. Except in the campsite analogy, I have not used the term to advance any proposition, and even the analogy is merely a suggestion, not an argument for a proposition.

The term was batted into my court by someone else. Perhaps, at that point I should have not even returned the serve, but instead stopped play to require that the person who introduced the term give a definition. (And anyone is still welcome to suggest their own definition.)

In other words, someone claims that something is arbitrary, and my response is (and perhaps I should have been clear that this is my response): I'm not sure what your philosophical sense of arbitrary is, and I don't have a philosophical demand to avoid arbitrariness, but I do have some sense of what you mean, and to the extent that I have a sense of it and have some sympathy for your concern, I offer an analogy with an everyday sense of 'arbitrary' that provides some entry way into further thinking on the subject, and if you would, please tell me more about your own philosophical sense of the term.

As to the everyday sense per the example I gave, I don't see how one would require more explanation than I gave in my previous post. At the campsite we chose a certain stick, but there was nothing distinguished about that stick that made it particularly more suitable for a measuring rod than many another other stick lying around the area. For that matter, we could have apprised a few sticks and by a series of coin toss elimination contests chosen the winner. And in that scenario, the decision for a particular stick would include also randomness, which is a concept close to arbitrariness.

A dictionary definition would suffice. Among many senses listed are:

* based on or determined by individual preference or convenience rather than by necessity or the intrinsic nature of something. [I might recast that as: based on whim or momentary expedience rather than with regard to principles or systematic evaluation.] Dictionary examples given: <an arbitrary standard>, <take any arbitrary positive number>, <arbitrary division of historical studies into watertight compartments>

* existing or coming about seemingly at random or by chance or as a capricious and unreasonable act of will [I'd change 'and' to 'or'.]

I don't have a mathematical definition, again, since I'm not claiming there even is a mathematical sense.

And, as I understand, Objectivists are concerned that concepts are formed with regard to essential properties of entities and that concepts not formed in this way are arbitrary. And I mentioned some of the questions I have about this.

You wrote, "How can one think about a subject without knowing what the terms/ideas refer to in reality?"

Indeed, and so I do understand the everyday sense of 'arbitrary' and welcome anyone to assist my understanding of an Objectivist definition of the philosophical sense.

And, to be clear, you first made explicit the notion in this thread that there is or might be a philosophical sense for the term. So, by my referring to this, I am deferring to you that there is or might be such a sense, which you are welcome to provide.

P.S. In an earlier post you commented that your not being a mathematician disqualifies you from commenting on the subject of this thread. For the record, as far as I am concerned, no one is disqualified by such a reason, though, of course, it is each our own prerogative to disqualify ourselves if we wish to.

[PhilE.Stein]

Note: Above post is addressed to Burgess Lauglin, as Capitalism Forever's post came between.

[PhilE.Stein]

Capitalism Forever,

Your formalization is a definition that says what it is for the cardinality of a set to be equal to the number two. And your formalization is equivalent to that of set theory in that regard. In other words, whatever definition a set theory will give, it will be a theorem of set theory that a set has cardinality equal to the number two if and only if <fill in your definiens here>. But your formalization does not give a definition of the number two itself.

Once you understand the Objectivist view of abstractions, you'll see that a formal system need not capture everything about reality in order to describe reality correctly.[right][snapback]13429[/snapback][/right]

Yes, and this follows not just from Objectivism, but from the very notion of abstraction. The point I had intended, and can now make more clear, is not that mathematics might fail to capture everything about reality, but rather that mathematics might fail to capture what is needed in a way that Objectivism would deem a representation of reality. In other words, the formulations might not have the kind of direct "counting apples" methodology that you described, but instead provide a representation that's more indirect and with rather oddly conceived entities, but still, in the end coming through to provide a formulation that provides equivalences with the basic concepts of counting apples, as seen, for example, in that your definition of the cardinality of a set being equal to the number two is equivalent with the set theory formulation of this, though set theory first went through some somewhat odd steps to arrive at this equivalence.

The reason I suggest that mathematics might not be able to achieve the direct method you've suggested is formalization. A language such as English allows us to convey things like "If I had an apple but not another, then it would be false that I have two apples." But some folks want to put that in a formal language, since we want to remove the clutter of talking about people having apples when the abstraction from people and apples is the goal, and we don't want to be subject to ambiguity that may arise in a natural language, and we want to reduce our language to the fewest primitive symbols possible, so that our hierarchy of definitions and theorems starts with as few presumptions as possible and we can trace back to notions that are simple and fundamental.

But, as I'm suggesting, the catch is, that do this, without all the expressive power of natural language, we make have to take curvy routes in our journey of abstraction conveyed only in formal symbols. And in these curvy routes we may find ourselves with entities and formulations that don't correspond to our empirical experience. However, it is hoped, these entities and formulations don't contradict our empirical experience. They are more like abstract artifacts that are used to finally arrive at the formulas that do correspond to empirical experience.

In your example of the plane having two engines, we have no difference regarding taking a set theoretic definition of 'two' literally. The set theoretic definition is not intended for people to do things as basic as counting and making basic calculations.

"The meaning of the word within the formal section should be the same as outside it.[right][snapback]13429[/snapback][/right]

But, as I mentioned, for the particular word 'set', that expression does not even occur in the formal language. Moreover, in formal languages, there are only expressions that occur only in the formal language. Also, formal languages are carefully divided between syntax and semantics. A definition in a formal language is strictly syntactical, and does not stipulate meaning except as the formal theory itself limits what models are models of the theory. Some realist mathematicians take these models as realities, but other mathematicians regard the models as not requiring ontological commitment.

Granted, informally, mathematicians freely words such as 'set' to reference things, or ideas, or concepts ('concepts' here not necessarily in an Objectivist sense) and even begin textbooks on the subject with such talk. So, if that usage does not agree with what you consider proper use of words such as 'set', then if one were to be gracious, one would concede by admitting that mathematicians should use their own invented locutions, such as, say, 'zet' instead of 'set'.

[...] the meaning of "where A, B are sets" is "where A and B refer to mental groupings of objects, arising either as enumerations or as combinations of a base class and a predicate."[right][snapback]13429[/snapback][/right]

That's fine, except how to formalize it? Or, if not formalized, then how do you check whether symbols on a page refer to mental groupings of objects arising either as enumerations or as combination of a base class and a predicate.

Also, terms that require definition are 'grouping', 'enumeration', 'combination', 'base', 'class', and 'predicate'. My point is not to indiscriminately make one chase for definitions of every word used in a conversation. But, at the very very crucial point of setting up a definition of perhaps the most fundamental term of mathematics, one does need to know the definitions of the terms used to define said term. Or, if the terms used to define are to be primitives, then one needs axioms for them.

You did give definitions of 'grouping', 'class', and 'predicate', but these definitions made use of terms that require definition. For example, you defined 'grouping' in terms of 'number' and 'predicates' in terms of 'attributes'. Eventually, we need to get to your primitive terms, and know your axioms for them. In everyday context one wouldn't make such demands for definition upon definition, but mathematics does require that we know what are our primitive terms.

Essentiality is a criterion for what to include in a definition of a concept.[right][snapback]13429[/snapback][/right]

I am interested in knowing more about your notion of essentiality.

The predicate should be applicable to the base class (you can use the predicate "red" with apples, but you cannot use it with air molecules, since air molecules do not have a color) and decidable for every instance of the base class.[right][snapback]13429[/snapback][/right]

Set theory can easily accommodate the criterion of applicability. Often mathematicians say things like "n is even if and only if n is a multiple of 2" as this is understood to apply only to natural numbers. But the definition could be over all objects by saying "n is even if and only if n is a natural number and n is a multiple of 2."

However, your criterion of decidability is a huge one. You'd have to give a decision procedure, and we'd have to know that these mathematical predicates are indeed decidable given your decision procedure. I am stating this informally, but it seems to be that even for just arithmetic, there can be no such decision procedure and that there will always be undecidable predicates. I've put that very very informally, so my statement may withstand sharpening. Nevertheless, it seems you've obligated mathematics to an impossible criterion.

It makes the infamous "all sets that do not include themselves" invalid as a set specification, because the predicate is undecidable for the set being defined due to circularity.[right][snapback]13429[/snapback][/right]

Roughly stated, set theory does avoid the circularity by not allowing the predicate to contain reference to the variable that stands for the defined set. However, this is much different from a criterion that predicates be decidable. And, as I mentioned, I don't know how you'll be able to ensure decidability in the general case.

An Objectivist mathematician does not need any such derivation to know that the set of natural numbers exists. He knows that counts exist in reality; he knows that they are natural numbers; he knows that he can mentally group natural numbers and that a mental grouping specified by a base class (and, in this case, a vacuously true predicate) is a set--therefore, he knows that the set of natural numbers exists.[right][snapback]13429[/snapback][/right]

But does an Objectivist know that we can mentally group an infinite number of things? If so, that's fine. But the set of natural numbers is an infinite set. If that's acceptable to Objectivism, then that's a point of agreement with set theory.

Also, a set theorist can make the same mental grouping, but the point of the set theoretic derivation is not to prove that such a mental grouping can be made, but rather to provide a completely formal "version" of this mental grouping in terms only of primitives and axioms.

I understand an implication being vacuously true, but I don't know what you have in mind, in this particular case, by "vacuously true predicate".