Axiom Of Choice?

[ewv]

Eric: Mathematical existence is a tricky subject.  I believe that most mathematicians don't really think about what they mean when they say "there exists". To them, it is just part of, and therefore gets its meaning from, the mathematical formalism.  This is obviously backwards.  Meaning must come first.

Mathematicians may not often think about the philosophical meaning of their concepts, but they know very well what they are doing and what it means technically to say something exists. Philosophically, they tend to err in practice on the side of Platonism (while espousing Formalism on "Sundays"), but to say they are manipulating empty mathematical formalism without regard to meaning is an astounding accusation against an entire profession.

Mathematics is full of so-called "pure existence" or "ideal existence" theorems, i.e., theorems that assert the existence of some object without giving the slightest clue as to how to find it. I believe that such theorems should be distinguished from ones which actually construct the object in question, since there is an obvious difference in pragmatic content. For example, if I need to find a root of an equation, a theorem that asserts the "pure" existence of a solution is of very little value compared to a theorem that constructively finds the root.

The two types are distinguished; who are you debating with?

Everyone knows the value of knowing a solution to an equation as opposed to only knowing that one exists. But taking a broader view within the science, as opposed to someone who only wants a single answer, there is enormous value to knowing the general nature of what exists as a "solution" to a whole class of problems, in advance of knowing a particular solution to a particular equation, for several reasons. One is the obvious value of knowing that it is worthwhile, and not futile, to search for a particular "solution" in any specific case. More broadly, general existence proofs tell you what exists in terms of its properties, which is extremely useful. Mathematics is typically concerned with whether the solution is unique and more about its character. If you know there may be more than one solution and you have only found one, you may not have the solution to your actual problem. Sometimes certain general properties are all you need to know for some physical problem, without knowing the exact solution. And the general properties can be enormously useful in telling you how to find a particular solution.

For example, it can be important to know how many solutions there are to a polynomial equation, whether they are real or complex, the sign, or the presence of complex conjugates. Knowing these characteristics in general is extremely useful for solving particular equations. If you are concerned with differential equations the degree of continuity and the stability are extremely valuable for characterizing whole classes of problems and provides important information on what to look for and how to do it in specific cases. And knowing the properties of the solution, such as the degree of continuity and how stable it is tells you a lot about what kind of numerical algorithms you can use to find it and what kind of accuracy you can expect at what cost.

Yet, today, these two different concepts are both lumped into the mathematicians' "there exists". If any distinction is maintained, it is that pure existence proofs are generally considered more "elegant" than constructive proofs - "elegant" apparently meaning less useful.

A non-constructive existence proof may be elegant if establishes something of importance with great conceptual economy; a constructive proof may do that, too, but usually requires more details of the construction and may therefore be regarded as less "elegant". That does not mean that it is "less useful". Proofs that establish an example of existence or an algorithm are generally regarded as more useful for obvious reasons.

I personally don't think that "pure existence" is existence. Existence is identity, and absolutely nothing has been identified in a pure existence proof. This is why I deny that the set E above exists: the Axiom of Choice asserts only pure existence.

There is no such thing as "pure existence" apart from identity and no mathematical existence proof claims to establish the existence of anything without identity. On the contrary there is nothing to prove if no property, i.e., aspect of identity, is specified. Non-constructive existence proofs establish abstract assertions of existence. It is analogous to a theoretical chemist establishing that some new element with certain properties must exist, without having a physical example of it in hand. That is not a meaningless existence devoid of identity.

The Axiom of Choice specifies the existence of a set whose members are related to other sets, without specifying which members. That level of abstraction does not violate identity; the particular members are an omitted measurement. At the level of abstraction of many proofs, you don't need to be more specific than that.

This is not to say that "pure existence" theorems are useless. They tell us, for example, to not bother looking for a counterexample and to focus our efforts on constructing a real solution.

So they are "useful" but meaningless because they have no "identity"? If they don't establish that something exists then what is it that you don't look for a counterexample of? You must have identified something that is the subject of attention. If you know that something must exist that you can try to find, you know it actually exists, and you know enough about the "it" to know what you are talking about.

[stephen_speicher]

I agree with most everything that emv said about the law of excluded middle. One comment, though, on a side issue.

Nevertheless, his critique generated a lot of philosophical sympathy from mathematicians.  Even Hilbert took it seriously.  He rejected Brouwer's proscription, but Brouwer's critique was the motive for Hilbert's formulation of what became known as Formalism in the foundations of mathematics ...

That is the traditional view regarding Brouwer and Hilbert -- one that I held until recently -- but I think there is strong evidence that Hilbert's whole program was more self-motivated, philosophically, and not really as a response to Brouwer. If anything, amazingly enough, there is also some evidence that Cantor may have laid the seeds for Hilbert's program. If anyone is interested, here are two good references as a starting point for exploring this thesis.

"Hilbert's Philosophy of Mathematics," M. Giaquinto, The British Journal for the Philosophy of Science, Vol. 34, No. 2, pp. 119-132, June 1983.

"Hilbert's Programs: 1917-1922," W. Sieg, The Bulletin of Symbolic Logic, Vol. 5, No. 1, pp. 1-44, March 1999.

[ewv]

Hilbert embarked on a number of ambitious projects to establish formal mathematical axioms for different branches of mathematics and physics, beginning with his early comprehensive revision of the axioms for Euclidean Geometry. His approach to Formalism as his foundation for mathematics as such was intended to cover several major issues besides the validation of the use of the mathematical infinite; he may very well have followed such a comprehensive path even without Brouwer. I can't comment on the historical sequence further here because I'm away from my references (and haven't seen the articles mentioned by Stephen), but I do think it's important to note the historical connection between Brouwer's challenge regarding the infinite in particular and Hilbert's attempt to deal with it without limiting the applicability of the Law of the Excluded Middle. The source of the controversy was not the Law of the Excluded Middle itself, but how to handle the mathematically infinite. Brouwer's approach was widely regarded as well-motivated but over-restrictive in his limitations on the use of the Law of the Excluded Middle for the infinite, leading to other kinds of approaches.

What was needed was a comprehensive, correct philosophical accounting of the nature of mathematical abstraction and systems. Without a correct epistemological solution to the problem of universals in general, that was not feasible. The mathematical infinite can now be clearly understood and explained with Objectivist epistemology, clearly illustrated using several technical advances especially from 19th century mathematics. The technical foundations of mathematics and the epistemologically proper use of set theory is a broader issue yet to be resolved.

[stephen_speicher]

I can't comment on the historical sequence further here because I'm away from my references (and haven't seen the articles mentioned by Stephen), but I do think it's important to note the historical connection between Brouwer's challenge regarding the infinite in particular and Hilbert's attempt to deal with it without limiting the applicability of the Law of the Excluded Middle.  The source of the controversy was not the Law of the Excluded Middle itself, but how to handle the mathematically infinite.

Two brief comments here.

First, considering the time prior to Brouwer, it was commonly thought that Hilbert abandoned his foundational work between 1904 and (at least) 1917. For instance, one finds in Morris Kline's magnificent tome Mathematical Thought from Ancient to Modern Tines, p. 1204, Oxford University Press, 1972, this comment about Hilbert and foundational work:

"Hilbert presented one paper on his views at the International Congress of 1904. He did no more on this subject for fifteen years; then, moved by the desire to answer the intuitionist's criticisms ..."

It is true that one cannot find papers during that period, but more recent investigation has revealed, through the content of lectures and other sources, that indeed Hilbert was actually quite active during those years on foundational issues.

Second, just so that there is no misunderstanding on the part of others, if one were to identify the core element of Hilbert's foundational program, it was to establish the basis for all set mathematics by a reduction to finitism. In essence, Hilbert agreed with Aristotle that infinity did not refer to anything actual in physical reality, so he attempted to rework the whole system of infinitistic mathematics to give a finitistic consistency proof for it all. I do not think that Brouwer ever really understood Hilbert's work, as Hilbert intended to give meaning to infinitistic formulas via his finitistic reasoning and mathematics.

[Eric]

Brouwer's intuitionist logic rejects the law of excluded middle....  Aristotle got it right in the first place, and I really have no interest in any sort of discussion where the law of excluded middle is cast aside.

Neither I nor Brouwer completely reject LEM. Of course Aristotle got it right: he developed and applied it in finite situations. Would he approve of its use when dealing with the infinite and with existence, as is done by modern mathematicians? I doubt it.

I agree with Hilbert who was quoted as saying "Forbidding a mathematician to make use of of the principle of excluded middle is like forbidding an astronomer his telescope or a boxer the use of his fists."

Hilbert's statement is hyperbole. Brouwer showed that mathematics - serious mathematics - can be done while restricting the use of LEM to finite situations. Later, in the 60's, Errett Bishop, in his book Foundations of Constructive Analysis, single-handedly developed most of analysis in a constructive framework. And, because LEM isn't used to create mathematical objects ex nihilo, the resulting work is more meaningful (and useful!) than classical analysis. When Bishop proves the existence of a mathematical object, I believe him. When Banach and Tarski "prove" that a sphere can be decomposed into a finite number of pieces and reassembled into two spheres identical to the original, I laugh at the absurdity.

[Eric]

The law of the Excluded Middle as formulated and advocated by Aristotle was not and is not limited to the finite

I'm not aware of Aristotle ever advocating using LEM with the infinite. Could you point me to where he does?

Brouwer didn't "show" there are limits to the Law of the Excluded Middle, nor in particular did he restrict it to the finite.

To Brouwer (and me) a mathematical object exists if and only if it can be constructed. Thus, assuming that an object doesn't exist and deriving a contradiction does not allow one to use LEM to conclude that the object exists. A contradiction is not a substitute for a construction.

Brouwer did indeed limit the use of LEM to propositions whose truth or falsity could be decided in a finite number of steps.

that does not mean that well-defined mathematical concepts like the well-ordering of the reals are meaningless or equivalent to mystical assertions of undefined beings.

The statement "A well ordering of the Reals exists" is arbitrary mysticism. The well ordering is conjured up by the axiom of choice. It has no referent in reality. It has no use in reality. No one has ever constructed one. No one ever will.

[ewv]

Eric: I'm not aware of Aristotle ever advocating using LEM with the infinite. Could you point me to where he does?

Not while I'm away from my references. But consider the next few points, which may clear it up when you see more of what I mean.

To Brouwer (and me) a mathematical object exists if and only if it can be constructed. Thus, assuming that an object doesn't exist and deriving a contradiction does not allow one to use LEM to conclude that the object exists. A contradiction is not a substitute for a construction.

Reduction ad Absurdum is not a substitute for a constructive proof in that they do not provide the same kind of insights, but both are valid means of proof. Mathematics is not directly about physical "things" in reality (either this one or some alleged Platonic realm). At its very base, even the concepts of numbers require the explicit idea of an abstract unit and are higher level abstractions than concepts of things like marbles, rocks, etc. used as examples of how to grasp counting and simple arithmetic. Mathematical concepts are very high level, abstract concepts of method. Most of these issues of mathematical "existence" should be thought of as the discovery or identification of relationships implicit in already validated concepts, not identification or literal construction, one by one, of "things". Be careful of how you think of the term "mathematical entity".

The Law of the Excluded Middle applies to any statements that are meaningful, and is used all the time with open-ended concepts. Brouwer missed the point when he proposed limiting the Law of the Excluded Middle instead of directly going after the proper criteria for valid concepts involving the mathematical infinite. (He couldn't have successfully done that philosophically himself because he was a Kantian.) His mathematical idea of restricting the infinite to the constructible infinite was in the right direction because it captures the idea of the infinite as a potential with some specific means of proceeding, but his approach in general turned out to be too limiting.

Brouwer did indeed limit the use of LEM to propositions whose truth or falsity could be decided in a finite number of steps.

Don't equate the mathematical infinite, e.g. in the concept of the integers, with the number of steps in a logical argument. An argument without a specific number of steps because it never stops has no specific identity itself and so is not an argument at all and can therefore establish nothing. But you can certainly make "finite" mathematical arguments about the integers, for example, as open-ended, i.e., an "infinite set". Even Brouwer acknowledged and advocated that. Such arguments include the use of the Law of the Excluded Middle and mathematical induction, i.e., based on recursion. The latter is itself a "finite argument" and is an interesting topic philosophically which is sometimes misunderstood as meaning an actual infinite.

"that does not mean that well-defined mathematical concepts like the well-ordering of the reals are meaningless or equivalent to mystical assertions of undefined beings."  The statement "A well ordering of the Reals exists" is arbitrary mysticism. The well ordering is conjured up by the axiom of choice. It has no referent in reality. It has no use in reality. No one has ever constructed one. No one ever will.

Whether or not you think the statement is true for some ordering or that the Axiom of Choice is improperly used for an incomprehensible use of infinity, the meaning of the concept of "well ordering" is clear and you can meaningfully ask the question. This is not mysticism: Again, mathematics is not about physical "things" in reality; the concepts are very high level abstract concepts of method and we are dealing with the discovery or identification of relationships implicit in already validated concepts. If improperly done it can be subjectivism, but it is not mysticism. If someone claimed there is a well-order of the reals as an actual infinity in some kind of Platonic realm, that would be mysticism.

[punk]

Brouwer rejects the law of the excluded middle because he is giving a non-standard interpretation of "true" in a mathematical context.

Brouwer is reading "true" more in the direction of "it is provable that" (not exactly right, but good enough). So given a statement A, Brouwer reads A or -A along the lines of:

it is provable that A or it is provable that -A.

Since it is possible that neither is provable, the law of the excluded middle does not apply.

I am resurrecting this because it seemed to stop at the crucial philosophical question: What does it mean to be "true" in mathematics? One's view of the Axiom of Choice really rests on this question.

The conventional notion of "true" in mathematics is basically Platonic in nature. That is a mathematical statement is true just in case that is the way the universe is (or more mythologically, because when we go to Plato's heaven, the statement is there, so it is "true").

Brouwer rejects this in favor of a more functional definition of truth.

[jrs]

In his second message of Oct 23, 2004, Eric said "The statement 'A well ordering of the Reals exists' is arbitrary mysticism. The well ordering is conjured up by the axiom of choice. It has no referent in reality. It has no use in reality. No one has ever constructed one. No one ever will.".

Given a model of Zermelo-Fraenkel set theory, one can form an inner model by limiting oneself to the class, L, of constructible sets. Kurt Goedel showed that L satisfies the axioms of Zermelo-Fraenkel set theory and also "V=L" (an axiom saying that all sets are constructible).

"V=L" implies both the Axiom of Choice and the Generalized Continuum Hypothesis. In fact, the entire Universe of the Model, that is the class L, can be well ordered by an explicit formula. Restricting the application of this to the set of real numbers gives an explicit well ordering of the (constructible) real numbers.

See "Lecture Notes in Mathematics" #223 "Models of ZF-Set Theory" by Ulrich Felgner, published by Springer-Verlag in 1971.

Essentially, one gives a procedure for constructing all constructible sets one at a time. Each ordinal number is interpreted as a code for constructing a set; and each set is the result of such a construction. The real numbers (which are sets) are then ordered by the ordinals that code for them.

[jrs]

Randrew:

You said "The problem with the Axiom of Choice is that, using it, one can prove the infamous Banach-Tarski Paradox which states (actually, implies) that one can take a sphere, cut it into six parts, and rearrange the parts to form a new sphere with double the volume of the original (without creating any new empty space inside the object.)".

[i remember the statement of the paradox differently, but I never studied it.] This is a reflection of the fact that there are non-measurable sets, that is, sets which do not have a precise volume.

This is not a fault of the Axiom of Choice which is merely involved as a convenient way of establishing the existence of non-measurable sets with the necessary properties.

The Banach-Tarski paradox is only a problem as long as one persists in trying to treat sets which have no precise volume as if they did have a precise volume.

Russell's paradox was an actual contradiction in Frege's development of Cantor's set theory. Thus that naive set theory was destroyed.

The Zermelo-Fraenkel Set Theory was constructed specifically to avoid Russell's paradox. As far as we know now, it is consistent.

ashleyisachild:

Mathematicians do NOT consider the Axiom of Choice to be contradictory!

You asked "What is the standard ordering [of the reals]?".

As you know, the standard ordering of the integers goes like this:

... < -3 < -2 < -1 < 0 < +1 < +2 < +3 < ...

If you divide them by some positive integer such as twenty, you get:

... < -3/20 < -1/10 < -1/20 < 0 < +1/20 < +1/10 < +3/20 < ...

If you merge all such orderings together, you get the standard ordering of the rational numbers. In that ordering,

(P/Q) < (R/S) means (P*S) < (R*Q) in the standard ordering of the integers, provided that P,Q,R,S are integers and Q,S are positive.

Imagine cutting the rational numbers in two with knife in such a way that there are rationals both on the left and on the right and there is no largest rational on the left (smaller) side of the cut. These cuts are the real numbers. If there is a smallest rational on the right side, it is considered to be equal to the real number.

The standard ordering of the real numbers is ordering them by how far to the right the cut is in the rationals. So the square root of two is less than pi, because the square root of two is less than (say) +5/2 which is a rational number and +5/2 is less than pi.

Mathematicians are well aware that the standard ordering of the reals is not a well ordering. The Axiom of Choice implies that there are well orderings of the reals, but it does not specify any particular one.

You asked "How does it do that [assert existence without identity]? But doesn't all of math do that?".

Mathematics is based on concepts in which all characteristics of the units (other than the fact that they are units) have been omitted.

Eric:

I remember from when I studied logic and foundations that there was something called the "no counter-example interpretation" of sentences in analysis. The idea was that one could construct a series of generalized recursive functionals associated with each step of a proof. If someone presented a supposed counter-example to the resulting theorem, then the functionals associated with it would generate a case where the counter-example did not work.

So in number theory at least, it should be possible in principle to convert any proof of mere existence into an actual derivation of a number which has the required property.

In practice, this calculation would often be impossibly difficult.

[Hal]

I remember from when I studied logic and foundations that there was something called the "no counter-example interpretation" of sentences in analysis.  The idea was that one could construct a series of generalized recursive functionals associated with each step of a proof.  If someone presented a supposed counter-example to the resulting theorem, then the functionals associated with it would generate a case where the counter-example did not work.

So in number theory at least, it should be possible in principle to convert any proof of mere existence into an actual derivation of a number which has the required property.

In practice, this calculation would often be impossibly difficult.

You mean you can prove that a constructive proof exists without actually being able to construct it?

I found your post interesting, thanks.

[jrs]

Hal:

You asked "You mean you can prove that a constructive proof exists without actually being able to construct it?".

Ouch!

Also the well-ordering of the real numbers in my earlier message is not computable even in principle.

[Franklin]

Okay, I believe that I may be the friend that she is referring to, and I think my case has been misrepresented by the fact that Ashley doesn't have---no offense intended---the proper mathematical background to make my point. So here it is:

There is a school of logic known as intuitionism, which is a member---so to speak---of another school of logic known as constructivism. Constructivism eliminates the law of the excluded middle (except for equality, where its use is a therom rather than an axiom). Intuitionism takes this a step farther and eliminates all mathematical entities that cannot be directly constructed. It is a school of mathematical philosophy rooted in Kant's notion of mathematics as being a construction of the mind that has no correllary in external reality. An example of its ridiculousness is its rejection of the axiom of choice because this axiom has been shown to be equivalent to Zorn's lemma, and a well-ordering frequently cannot be concretely portrayed. [As an aside I should say that intuitionism, as a formal method and not as a philosophy of math as a whole, does seem to have value in computer logic because computers can't think abstractly and must calculate their answers to logical questions, but this fortuitious use is more an accident than a justification for the ideas behind intuitionism]

Now here is my point, Peikoff's notion of the arbitrary claim---at least as most objectivists I have meet misunderstand it----seems disturbingly similar to this particular school of logic, which disturbs me because anything based in a Kantian epistemology is highly unlikely to be correct.

I will give you an example of a debate I had with another objectivist over this issue. We were at a friend's apartment reading OPAR and discussing the chapter on logic. In response to Peikoff's section on arbitrary claims, one of the people present said roughly the following:

'So if I say that there is a cat in the closet, that is an arbitrary claim since I don't have any direct evidence either way regarding that." I said that he was clearly misunderstanding the idea and that "No, arbitary claims refer only to the CATEGORY and not to particular utterances that are grammatical and that consist of members of categories whose existance is well verified; thus, if I say God is behind the door, that is arbitary for there is no evidence that allows for the creation of such a category as God, but "cat" and "closet" and "in" are all well defined and thus not arbitrary claims" I then tried to explain how his reasoning would effect notions of proof in math and physics, but he would not listen. So while I claimed that one could say that "the cat is behind the closet is either true or false" he said that it is neither true nor false. This leads him into the school of intuitionism which would put---if he were right---objectivism at odds with classical Aristotlean logic and first and second order logic.

By the way, here is an interesting example of a proof using intuitionistic methods. I will draw you attention to how clumsy it is just to define the reals (in comparison to the Dedekind cut definition).

http://cs.nyu.edu/pipermail/fom/2000-July/004199.html

Edited August 8, 2005 by Franklin

[Hal]

'So if I say that there is a cat in the closet, that is an arbitrary claim since I don't have any direct evidence either way regarding that." I said that he was clearly misunderstanding the idea and that "No, arbitary claims refer only to the CATEGORY and not to particular utterances that are grammatical and that consist of members of categories whose existance is well verified; thus, if I say God is behind the door, that is arbitary for there is no evidence that allows for the creation of such a category as God, but "cat" and "closet" and "in" are all well defined and thus not arbitrary claims" I then tried to explain how his reasoning would effect notions of proof in math and physics, but he would not listen. So while I claimed that one could say that "the cat is behind the closet is either true or false" he said that it is neither true nor false. This leads him into the school of intuitionism which would put---if he were right---objectivism at odds with classical Aristotlean logic and first and second order logic.

What about "there is an elephant in the closet", or "there is cat on the moon (presumably dead)"? Does your metalogic contain a decision procedure for determining whether a statement is arbitrary and hence outwith the formalisation?

I partially agree with your friend - talking about 'arbitrary statements having no truth value' is fine in informal logic, but if you want to construct some kind of calculus, youre going to have to be far more precise than this. Theres nothing wrong with declaring that some statements dont have a truth value - there was a big debate in the mid 20th century between Russell and Strawson concerning whether statements involving non-existent entities (eg "the king of France is bald", "the golden mountain is golden") should be labelled 'false', or just classed as meaningless without getting a truth value. But you need to have some objective way of determining whether a given statement is arbitrary. The fact that there doesnt even seem to be agreement on whether 'there is a cat in the closet' is arbitrary doesnt look very promising (I'd personally say the arbitrariness of this statement depends entirely on context - in a household with 9 cats this statement wouldnt be arbitrary, but in an animal-free house with all doors and windows locked, it would be).

Edited August 9, 2005 by Hal

[DavidOdden]

I partially agree with your friend - talking about 'arbitrary statements having no truth value' is fine in informal logic, but if you want to construct some kind of calculus, youre going to have to be far more precise than this.

You can start by recalling that an arbitrary statement is one lacking evidence. To quote OPAR 164, "An arbitrary claim is one for which there is no evidence, either perceptual or conceptual. It is a brazen assertion, based neither on direct observation nor on any attempted logical inference therefrom." That means that an arbitrary statement is not an observation of fact, and it also cannot be reduced to observation of fact. As such, the statement is outside the domain of logic.

The fact that there doesnt even seem to be agreement on whether 'there is a cat in the closet' is arbitrary doesnt look very promising (I'd personally say the arbitrariness of this statement depends entirely on context - in a household with 9 cats this statement wouldnt be arbitrary, but in an animal-free house with all doors and windows locked, it would be).

But that's the point. It's arbitrary for me to say that you're bald, but not arbitrary for me to say that you're English. It's about evidence.

[Hal]

You can start by recalling that an arbitrary statement is one lacking evidence.

I think that once we move away from trivial examples ("gremlins reading Hegel on mars"), it can become difficult to say whether there is evidence or not. I've had several arguments on this forum where others have claimed that there is no evidence that computers can be conscious, or that randomness exists in quantum physics, yet have point-blank refused to describe what they would consider to be acceptable evidence for these claims. In a lot (most?) of debates I've encountered, one side generally asserts that certain facts constitute evidence for their position, whereas the other side denies these facts are indeed evidence.

None of this is especially problematic - disagreements will always occur and in some cases, a proper analysis will show that one side is flat out wrong in their belief that evidence does or doesnt exist. But we are talking about formal logic here, and it seems odd (to say the least) that extra-logical factors will prevent a wff from getting a truth value without having a clear and unambiguous decision procedure. It would cause nightmares for your proof theory at any right.

As such, the statement is outside the domain of logic

In all honesty, I think its just wrong to say that 'there are dragons on neptunes' is neither true nor false. Regardless of whether evidence exists, there are either dragons on Neptune or there arent. Denying this seems to be some kind of bizarre 'primacy of consciousness' approach, where reality is in a state of indeterminacy until we manage to find evidence and collapse the 'superposition'. 2500 years ago, noone had evidence that the Earth was round but this did not change the fact that the Earth was round, and that "the Earth is round" was a true statement. Any theory of truth which denies this seems to be pointlessly subjective.

I understand (and agree with) Peikoff's view that we shouldn't give significance to arbitrary assertions in debates and discussions (or indeed our own thoughts), but its just perverse to hold that they 'arent really true or false'. Along similar lines, I agree with the idea that knowledge requires some sort of cognitive content ('grasping the facts of reality'), but I think there are far better ways to make this point than to hold that a parrot saying 2+2=4 isnt really making a true statement. You could say "the parrot is making a true statement although it doesnt know the statement is true", or "the parrot is making sounds which form a true proposition in mathematics", or countless other things.

.But that's the point. It's arbitrary for me to say that you're bald, but not arbitrary for me to say that you're English. It's about evidence.

This is actually a good example of what I mean. Does knowing that I live in Britain constitute evidence that I'm English? (does knowing someone lives in America constitute evidence that he's Texan?). I'm not English btw.

And even though you have no evidence whether I'm bald, it is still true that I'm not bald. You could always say "Its true for you but not for me", but this would an example of what I meant by pointless subjectivism. It seems a lot more common-sensical to say "If DavidOdden and Hal both asserted that Hal wasnt bald, then both would be right but in DavidOdden's case it would just be a lucky guess because he had no evidence either way, whereas Hal had lots of evidence and hence possessed actual knowledge". Although again - what constitutes evidence? The vast majority of people arent bald, so in the absence of more specific factors, you would have strong evidence that I'm not bald based purely on statistical liklihood (and if p isnt arbitrary, then surely ¬p isnt either)

Edited August 9, 2005 by Hal

[DavidOdden]

I've had several arguments on this forum where others have claimed that there is no evidence that computers can be conscious, or that randomness exists in quantum physics, yet have point-blank refused to describe what they would consider to be acceptable evidence for these claims.

Let's take the randomness example, and suppose that you think it's true. Now, what is your evidence? (Note that I'm not going to start by describing acceptable evidence for that turkey, but I'm willing to listen to arguments that something is evidence, at least up until that evidence crumbles).

But we are talking about formal logic here, and it seems odd (to say the least) that extra-logical factors will prevent a wff from getting a truth value without having a clear and unambiguous decision procedure. It would cause nightmares for your proof theory at any right.

This is where certainty becomes relevant. A decent logic does not let improbable claims be taken to be true; indeed, my conjecture based on scan evidence (language too!) that you're English (Ach! A Scotsman!!) wasn't arbitrary, but it also can't be entered as a valid truth.

In all honesty, I think its just wrong to say that 'there are dragons on neptunes' is neither true nor false.

Well, you're just grousing about the wording. It describes a fact, or an anti-fact. Reality has this "is" / "isn't" character. But truth is epistemological: it's a particular relation between fact and a consciousness.

(Parrot stuff demain)

[DavidOdden]

Carrying on...

Along similar lines, I agree with the idea that knowledge requires some sort of cognitive content ('grasping the facts of reality'), but I think there are far better ways to make this point than to hold that a parrot saying 2+2=4 isnt really making a true statement. You could say "the parrot is making a true statement although it doesnt know the statement is true", or "the parrot is making sounds which form a true proposition in mathematics", or countless other things.

Clearly, the parrot isn't making a statement at all. The latter is almost correct. Now imagine that Smith, a monolingual Texan who has a brain fever and is hallucinating, utters "Ugh ugh blugh blugh ugh blugh blugh" as Schrödinger's pet cat walks by. By coincidence, this sequence of sounds resemble a particular proposition articulated in Walmatjari meaning "The cat is dead", to the point that speakers of Walmatjari agree that this is what Smith said, and furthermore that his command of Walmatjari is impeccable. Even worse, it also turns out that this string of sounds expresses the proposition "The cat is not dead" in Piraha. A real-world contradiction -- a proposition that is simultaneously true and false. For a less exotic example, the proposition "That's a bad coat" is simultaneously true and false, depending on who says it and who they're talking to.

In other words, it is a mistake to confuse linguistic form with cognitive content. A propositions is a mental state, an assertion about relations between existents, which may be externalised as a sentence. Sentences and propositions are not the same: propositions are true or false, and sentences are true or false depending on what proposition they express.

[Franklin]

What about "there is an elephant in the closet", or "there is cat on the moon (presumably dead)"? Does your metalogic contain a decision procedure for determining whether a statement is arbitrary and hence outwith the formalisation?

I partially agree with your friend - talking about 'arbitrary statements having no truth value' is fine in informal logic, but if you want to construct some kind of calculus, youre going to have to be far more precise than this. Theres nothing wrong with declaring that some statements dont have a truth value - there was a big debate in the mid 20th century between Russell and Strawson concerning whether statements involving non-existent entities (eg "the king of France is bald", "the golden mountain is golden") should be labelled 'false', or just classed as meaningless without getting a truth value. But you need to have some objective way of determining whether a given statement is arbitrary. The fact that there doesnt even seem to be agreement on whether 'there is a cat in the closet' is arbitrary doesnt look very promising (I'd personally say the arbitrariness of this statement depends entirely on context - in a household with 9 cats this statement wouldnt be arbitrary, but in an animal-free house with all doors and windows locked, it would be).

Yes, it does. Namely, you go to the moon and you see whether there is a God damned rat on the God damned moon. The formal procedure is not difficult at all. This is what is so frustrating about most objectivists' interpretation of what Peikoff said. Yes, there is a distinction between cognitive content and formalism, but formalism can be applied powerfully without "a meaner" as long as a process of evidence/science grounded abstraction went into it and it eventually reattaches itself to the human: As evidence, I give you the thing you are looking at right now: your computer.

If the man I spoke of had the correct interpretation of what Peikoff said (and maybe he did but that would make Peikoff somewhat wrong), the words you see on your computer screen would be meaningless. Afterall, your computer is dumber than a parrot.

Now I must qualify this, since obviously some of the meaning you see is generated by the computer and others by people, so please don't use that to make a rhetorical jab. But if your computer says "Boss file out autoexec" or whatever, that statement has meaning despite not having a "meaner". I agree that cognitive content should be distinguished from linguistic form, but this does not mean that linguistic form should be treated constructively or that the law of the excluded middle should be discarded.

As for the last point, yes the metalogic---for lack of a better term---does contain a verification procedure. Namely, it is the procedure of verifying the empirical existance of the categories being used. This is the same procedure that Hilbert suggested in his philosophy of mathematics---your terms either show themselves to be useful and self consistent means of predicting sense experience or of describing this experience. Rat, for example, is a pretty well delineated category, as is moon, as is on. There is empirical evidence for the existance of the subcategories even if there is not for the main category.

If you apply any other schema, all of human abstraction is cut from underneath our feet: We can't even posit hypothetical scenarios because every claim, not just every category, has to be concretely checked. If "my friend" were right, we could never break things down into hypotheticals because each hypothetical would be a meaningless, "arbitary" claim.

Edited August 12, 2005 by Franklin

[DavidOdden]

Now I must qualify this, since obviously some of the meaning you see is generated by the computer and others by people, so please don't use that to make a rhetorical jab.

What do you mean by saying that meaning is "generated" by a computer?

[Franklin]

I'm not aware of Aristotle ever advocating using LEM with the infinite.  Could you point me to where he does?

To Brouwer (and me) a mathematical object exists if and only if it can be constructed.  Thus, assuming that an object doesn't exist and deriving a contradiction does not allow one to use LEM to conclude that the object exists.  A contradiction is not a substitute for a construction.

Brouwer did indeed limit the use of LEM to propositions whose truth or falsity could be decided in a finite number of steps.

The statement "A well ordering of the Reals exists" is arbitrary mysticism.  The well ordering is conjured up by the axiom of choice.  It has no referent in reality.  It has no use in reality.  No one has ever constructed one.  No one ever will.

Bouwer was/is a follower of Kant, by the way. Look. intuitionism certainliy is a useful methodology, but it is not a useful philosophy of mathematics. A well ordering can clearly be placed on a set---it is not a mystical object. Since the set is a formalism, just imagine the formalism but in the correct order. Let's say you have a library of 20,000,000,000,000,000,000,000,000 out ouf place books. The fact that all of humankind working together could never put them in order does not mean that the idea of well ordered library of 20.000.000.000.000.000.000.000.000 books is inconsistent. In fact, if the first is possible, the latter must also be possible. In fact, the set could be arranged in any order.

[Franklin]

What do you mean by saying that meaning is "generated" by a computer?

Some of the words, if you dislike my use of the term meaning.

[punk]

Bouwer was/is a follower of Kant, by the way. Look. intuitionism certainliy is a useful methodology, but it is not a useful philosophy of mathematics. A well ordering can clearly be placed on a set---it is not a mystical object. Since the set is a formalism, just imagine the formalism but in the correct order. Let's say you have a library of 20,000,000,000,000,000,000,000,000 out ouf place books. The fact that all of humankind working together could never put them in order does not mean that the idea of well ordered library of 20.000.000.000.000.000.000.000.000 books is inconsistent. In fact, if the first is possible, the latter must also be possible. In fact, the set could be arranged in any order.

Brouwer would agree that a well-ordering can always be placed on a finite set. The criticism was never that it would take *too long*.

Brouwer's criticism comes up when dealing with *infinite* sets. The point is that we cannot just assume a well-ordering exists for an infinite set, we have to come up with a procedure which generates the well-ordering.

For finite sets and problems intuitionism and conventional logic agree 100%. The disagreements only come up when considering infinite sets and problems.

[softwareNerd]

Some of the words, if you dislike my use of the term meaning.

David emphasized the word "generated".

In one sense of the term, everything on your computer screen is generated by the computer. In another sense of the term, it is all generated by the manufacturer and programmers. So, when you say the following:

some ... is generated by the computer and others by people...

it appears that you are either equivocating on the word "generated", or using it in a specific sense which needs to be made clear.

[Franklin]

Brouwer would agree that a well-ordering can always be placed on a finite set.  The criticism was never that it would take *too long*.

Brouwer's criticism comes up when dealing with *infinite* sets.  The point is that we cannot just assume a well-ordering exists for an infinite set, we have to come up with a procedure which generates the well-ordering.

For finite sets and problems intuitionism and conventional logic agree 100%.  The disagreements only come up when considering infinite sets and problems.

I understand the distinction, but I still think that my analogy holds (and it was just an analogy). If you accept that a well ordering holds for a very large finite set, but not for an infinite one, what is your rationale for claiming the idea inconsistent in the second case since in neither case can the well ordering be concretely demonstrated. The notion of well ordering would be equally consistent in both cases: Mathematics is not---as Bouwer and the Kantians claim----the act of constructing ideas in the mind (that is only a part of the activity and is more an aspect of applied mathematics). Rather math is the activity of testing the consistency of the mathematical ideas. The idea of a well ordering is perfectly consistent. If it isn't, please demonstrate an inconsistency.

Though I accept---as follower of Hilbert---that math has a strong basis in empirical reality, that does not mean that mathematical ideas are the same as empirical reallity: A work of fiction can be consistent, for example, and even meaningful with respect to the real world without being instantiated as an external set of historical events.