Schrodinger's Cat

[Darrell Hougen]

I know I responded to this point already, but before I leave the topic, I want to point out that the solopsist view depends upon any number of stolen concepts. A stolen concept is a concept that was developed to cover some collection of facts (which ultimately can be traced back to perceptions) but is now being used in a manner in which the context has been dropped. That is, the antecedent facts that gave rise to the original concept are now being ignored.

Our posts crossed, but in fact I answered this point already in my previous post. That the solipsist has to use words from our loaded language which imply an independent reality, but that doesn't prove him wrong. The argument of the "stolen fallacy" is not valid. On another forum I wrote: "What is wrong with the idea of the stolen concept fallacy: even if someone uses a concept in a way that denies its genetic roots, this doesn't necessarily invalidate his use of the concept. It merely means that the concept is no longer the same as the original concept. This may be perfectly valid, for example nowadays the concept 'time' is no longer the same as the concept 'time' before 1905. On the other hand is it of course possible that the new concept is not valid, but the point is that you can't prove that by merely pointing out that it denies the validity of its genetic roots. The SCF in itself doesn't prove anything, it's therefore only a rhetorical device without any real meaning."

A stolen concept is more than just a stolen word. It is an attempt to use the meaning of the word while ignoring the its referents in reality. The shift in the meaning of the word time is valid because new referents for its meaning have been provided in the form of experimental evidence justifying the new usage. No such justification is possible for the concepts stolen by the solopsist.

The final coup de grace is the theft of the concept of logic. Properly, a statement is logical if it corresponds to reality.

Certainly not. A mathematical argument can be strictly logical without having any correspondence to reality.

This is true only in the most vacuous sense, in the sense that the mathematical symbols don't mean anything. In other words, it is meaningless to state that if A relates to B in some way, then B relates to C in some other way, if A, B and C cannot utlimately be related to some existents that relate in the manner hypothesized. Without referents, the statements have no conceptual content. They simply amount to formal symbol manipulation in which the "symbols" don't symbolize anything.

Darrell

Edited February 21, 2007 by Darrell Hougen

[Darrell Hougen]

In the Objectivist view, it is not even sensible to talk about the logical consistency of the solopsist view, because the ultimate test of logical consistency is correspondence to reality. If there is no reality, then no correspondence can be established and consistency is meaningless.

That is begging the question, as you start with demanding correspondence to reality, which is not a condition for logical consistency.

To be logically consistent means to be consistent with evidence or observation. Essentially, to arrive at a logical conclusion means to make a prediction about the existence of some piece of evidence. If that evidence exists, then the conclusion is confirmed, otherwise it must be rejected and the process of reasoning must be checked or modified. It may be that some concept used in the reasoning process was not thoroughly understood, for example.

What you are talking about is the way that logic is often described in mathematics, the process of starting with a set of assumptions and applying rules to deduce a conclusion. However, the mathematical version of logic is just an abstraction of the process of reasoning about reality. Moreover, it is only reasonable and its conclusions are only meaningful insofar as they can be related back to actual facts of reality.

It is no accident that mathematics deals with numbers and quantities and geometric shapes, because those are things that either exist or are similar to things that actually exist, e.g., a circular object. However, you are attempting to invert the process by claiming that mathematical reasoning is valid, a priori, i.e., in the absence of anything that exists. I'm saying that you're putting the cart before the horse.

Darrell

[Michael Stuart Kelly]

I am really enjoying the word, SOLOpsist.

(Is that a stolen concept?)

Michael

[Darrell Hougen]

I am really enjoying the word, SOLOpsist.

(Is that a stolen concept?)

It might be, though now that you mention it, I discovered that I've been misspelling the word this entire time. It should be "solipsist."

[Guyau]

Darrell, I had written:

"The infinite sequence of rational numbers given by the formula {[(nxn)-n]/[2(nxn)+n]} with n=2,3,4,5,6,7,8, . . . converges step by ever-more-tiny step to the rational number (1/2). [That is the sequence: (1/5), (2/7), (1/3), (4/11), (15/39), (42/105), (14/34), . . . ] Though we cannot witness each number in the infinite sequence, the sequence is a definite singular abstract entity on the rational line Q.

"The question of whether a line in physical space contains all the points and properties of the rational line Q would seem to be an empirical question. But if lines of physical space do have all the points and properties of Q, then the sequence above is not only a definite single entity in abstraction, but in concrete space."

Part of the question of whether lines of physical space contains all the points and properties of the rational line Q---and this is the part I meant by the example---is the question of whether a line segment of physical space contains the points and properties of a segment of Q. Any segment of Q contains the particular infinite sequence of ordered ratios specified above. At a distance from one (or the other) end of the segment, a distance one-fifth the length of the segment, the sequence begins. At the midpoint of the segment is the unique point of convergence of the sequence, which the infinite sequence never reaches though it gets ever closer.

Physics has not ruled out the possibility of physical line segments containing all the points and properties of a Q segment. Likewise for the existence in physical space of the further points and properties of the real line R. If these possibilities are someday shown not to obtain in physical space, that circumstance will be expained by physics and the mathematics that has been found more physically fitting.

If you allow that a physical particle such as an electron could be (as it is presently thought to be in physics) a true single point having zero extent, then surely it is possible for there to be points of physical space that are true points having zero extent. And if such extensionless points of physical space are not to be excluded from physics, why exclude the breadthless lines that can be traced by such a point or the depthless surfaces that can be traced with such a line? Couldn't these be among the very lines and sufaces traced in space by moving bodies?

Edited February 22, 2007 by Stephen Boydstun

[bmacwilliam]

In the Objectivist view, it is not even sensible to talk about the logical consistency of the solopsist view, because the ultimate test of logical consistency is correspondence to reality. If there is no reality, then no correspondence can be established and consistency is meaningless.

That is begging the question, as you start with demanding correspondence to reality, which is not a condition for logical consistency.

To be logically consistent means to be consistent with evidence or observation. Essentially, to arrive at a logical conclusion means to make a prediction about the existence of some piece of evidence. If that evidence exists, then the conclusion is confirmed, otherwise it must be rejected and the process of reasoning must be checked or modified. It may be that some concept used in the reasoning process was not thoroughly understood, for example.

What you are talking about is the way that logic is often described in mathematics, the process of starting with a set of assumptions and applying rules to deduce a conclusion. However, the mathematical version of logic is just an abstraction of the process of reasoning about reality. Moreover, it is only reasonable and its conclusions are only meaningful insofar as they can be related back to actual facts of reality.

It is no accident that mathematics deals with numbers and quantities and geometric shapes, because those are things that either exist or are similar to things that actually exist, e.g., a circular object. However, you are attempting to invert the process by claiming that mathematical reasoning is valid, a priori, i.e., in the absence of anything that exists. I'm saying that you're putting the cart before the horse.

Darrell

This is a big problem here. This is not logic. Logic is by definition formal and abstract. Logic is the abstract rules that governs thought and reasoning. The connection to evidence or existence or anything is purely incidental. The pure realist position I can't support.

I don't think it's possible to have an intelligent discussion with such widely divergent definitions of logic. Personally I think this approach is a clear example of thinking too small and using our natural, but limiting cognitive bias to limit thought. This is not to say that logic can't predict anything, and if a prediction turns out to be false then sure, maybe the logic is wrong, but logic is separate from evidence. I remember reading stuff on this by Putnam?? I think and I just don't buy it. I mean, by this definition, don't we have to dismiss all non-euclidean geometry?

"However, you are attempting to invert the process by claiming that mathematical reasoning is valid, a priori, i.e., in the absence of anything that exists. "

It MOST CERTAINLY IS valid. This dismissal of abstraction is not even worth arguing about to me. Logic includes both real and abstract concepts.

Bob

Edited February 22, 2007 by Bob_Mac

[bmacwilliam]

I guess I can expand this thought a little more...

Darrell's position on logic, and mathemetics it seems too, demands a connection to reality and lives or dies, means something or means nothing on this basis.

I reject this, and I knew I read about it some time ago and I found a reference.

Stephen Yarnop at MIT wrote (discussing Carnap's ideas):

"How can an external deployment of 'there are X's' mean anything when by definition it floats free of the rules whence alone meaning comes?"

There's more to this argument of course, but that's the crux of it.

So, for example the question of whether logic, numbers, and other mathematical concepts exists, or your downstream extension of this concept where their validity is judged externally (agreement with observation) is an INVALID position. It is an IMPOSSIBLE question.

Now, I do understand the argument against this where the analytic/synthetic distinction is fuzzy then the distinction between or ontological/empirical is also called into question perhaps. However, it seems that the internal/external distinction is sharper and the argument against is weak. Essentially, the rules of the mathematical framework for example, are special in the sense that they more bullet-proof analytic. In a sense they are the only reason why the question of say "Do numbers exist?" has any meaning at all in even a linguistic sense.

So, that's just a long-winded explanation of why I view Darrell's view on logic and math, not just wrong, but misguided in the broadest sense because I have concluded that the question itself is non-sensical. Logic is valid period as a result of it's rules, so is mathematics. It's validity relies solely on internal criteria.

Therefore I cannot argue/discuss questions that I fundamentally view as invalid.

I like the analogy of playing tag - it's funny but accurate. What would children say about discussing the concept of who is really "IT" external to the game of tag? They'd say the question was stupid. The questions of whether or not mathematical reasoning is subject to external scrutiny is the same damn question.

Bob

P.S. - I also wanted to add that maybe on the surface it looks like there is something unscientific about logic and mathematics if they are held 'above' or separate from external questions or connections. I do not believe this to true at all. The reason is that the framework itself - the rules - can be modified and tested in light of observations and in fact this is precisely how new scientific models progress.

Edited February 22, 2007 by Bob_Mac

[Brant Gaede]

The laws of logic are validated by the manipulation of content which can be entirely hypothetical. (Also, logical fallacies.) Then you use them in the real world. I assume the same is true of mathematics. This is interesting to me only if I am wrong about math, but not interesting enough to go out and become a mathematician so I can get into an argument about this. My symbolic understanding of math is that it is used to cut a path through an otherwise impenetrable jungle or jumble of things. It's a guide and a tool. A mathematician is like a painter. Does a painter paint just so others might see or that he might also see?

--Brant

[Darrell Hougen]

This is a big problem here. This is not logic. Logic is by definition formal and abstract. Logic is the abstract rules that governs thought and reasoning. The connection to evidence or existence or anything is purely incidental. The pure realist position I can't support.

First, I'm not supporting a realist position. you are. The Platonic realist states that ideal forms exist independent of observation which is exactly what you're claiming. In the realist position, there is no explanation for how these ideal forms come to be associated with observations. This leads to the notion that God puts the ideals in people's minds as they are unable to do it themselves. There is a nice discussion of the historical and philosophical reasons for this in OPAR.

Next, your definition of logic is one possible definition, but is not the most useful definition. The reason that mathematical reasoning, for example, appears useful is because it deals with numbers (among other things) and we know that numbers of things exist, e.g., two cows, three oranges, six billion people (give or take a few). But, imagine what would happen if we started with the symbols, "!@#$%^&*()_-+=~," made up some axioms governing them (e.g., if @ and #, then $) and then began to reason about them. It would be possible, but the excercise would be meaningless. So, I'm not stating that purely abstract, formal reasoning is not possible, simply that it is meaningless if there is no connection to reality. A computer (as such exist today) can do formal symbol manipulation, but cannot understand its existence or the existence of anything else.

BTW, you should go back and read what I wrote about Gödel's proof. Gödel proved that in any formal system of sufficient power, there are infinitely many true statements that cannot be proved from any finite set of axioms, no matter how numerous. So, you should ask yourself how the truth of such statements is to be proved. Clearly, it must be with reference to reality.

Darrell

[Rodney]

To be logically consistent means to be consistent with evidence or observation.

I have to disagree with Darrell on this, and I am sure AR would disagree also.

It is true that Ayn Rand defined logic as “the art of non-contradictory identification” rather than “the art of non-contradictory assertion.” But this is only because she wished to emphasize the proper usage of logic—that it is supposed to be applied to experience in order to arrive at the truth and not engaged in as an end in itself. The fact remains that a conclusion can be perfectly logical and yet wrong because it is based upon false premises. That is, if logical consistency leads to a conclusion known to be false, at least one of the premises must be wrong. And this is just what Rand meant, I think.

Mathematics is of course another matter, and my forthcoming essay on hypercomplex numbers will, I hope, show this.

[bmacwilliam]

"First, I'm not supporting a realist position. you are. The Platonic realist states that ideal forms exist independent of observation which is exactly what you're claiming."

Yes, I think I misused that term for sure. I don't know off hand what "ist" or "ism" your empirical logic/math connection implies, perhaps is doesn't matter. Maybe Objectivist is most appropriate. However, I am most certainly not a realist in the Platonic sense. I mentioned nothing about "ideals" and the idea always sounded silly to me. Maybe I could be considered a 'critical realist' I guess (or at least some of that stuff makes sense to me), but that's very different than Plato's ideas about ideals and Gods and whatnot..

"Next, your definition of logic is one possible definition, but is not the most useful definition."

I think it's the only definition that makes any sense at all. And remember, an argument about definitions as far as I have ever seen is always fruitless. Can't get a common ground to proceed. Your definition is useless to me because, as I have explained, it's nonsensical. Like starting a conversation with " OK, assume you walk 4km north of the north pole then...." It's not a valid definition. In order to argue anything, there must be SOME common ground upon which to proceed.

You must not have understood what I wrote though. You avoided my direct criticism regarding your math/logic to reality link.

"So, I'm not stating that purely abstract, formal reasoning is not possible, simply that it is meaningless if there is no connection to reality. "

Well, that's somewhat different than your original statement about logical conclusions:

"arrive at a logical conclusion means to make a prediction about the existence of some piece of evidence."

Nonetheless, my point is that testing the validity of logic and by extension Mathematics by an external connection to reality is a nonsensical thing to do. You have not attempted to discuss this.

I don't know how Godel fits into this, but I'll think about it...

Bob

[bmacwilliam]

To be logically consistent means to be consistent with evidence or observation.

I have to disagree with Darrell on this, and I am sure AR would disagree also.

It is true that Ayn Rand defined logic as “the art of non-contradictory identification” rather than “the art of non-contradictory assertion.” But this is only because she wished to emphasize the proper usage of logic—that it is supposed to be applied to experience in order to arrive at the truth and not engaged in as an end in itself. The fact remains that a conclusion can be perfectly logical and yet wrong because it is based upon false premises. That is, if logical consistency leads to a conclusion known to be false, at least one of the premises must be wrong. And this is just what Rand meant, I think.

Mathematics is of course another matter, and my forthcoming essay on hypercomplex numbers will, I hope, show this.

Aha... I see - good post.

"But this is only because she wished to emphasize the proper usage of logic—that it is supposed to be applied to experience in order to arrive at the truth and not engaged in as an end in itself"

This makes sense in a value-judgment context, but no more. Just like saying "I don't like chocolate". True perhaps, but in a narrow sense and with limited applicability.

Defining logic with the word "art" is more than a small warning sign I'd say.

Bob

[reason.on]

They simply amount to formal symbol manipulation in which the "symbols" don't symbolize anything.

I'm saying that you're putting the cart before the horse.

Exactly. And to put a slightly different spin on it, neither a "brain" nor a computer "calculates" any more so than does an abacus.

RCR

Edited February 22, 2007 by R. Christian Ross

[bmacwilliam]

They simply amount to formal symbol manipulation in which the "symbols" don't symbolize anything.

I'm saying that you're putting the cart before the horse.

Exactly. And to put a slightly different spin on it, neither a "brain" nor a computer "calculates" any more so than does an abacus.

RCR

I could agree with that perhaps, but that's not the point I'm disputing. The sentence BEFORE the quote you included was:

"However, you are attempting to invert the process by claiming that mathematical reasoning is valid, a priori, i.e., in the absence of anything that exists. I'm saying that you're putting the cart before the horse."

This is wrong. Mathematical reasoning is defined by it's own terms ALONE - not external. Asking for the reality link to validate it is nonsensical. Just like asking what being "IT" means when you're not playing tag. The question is "illegal". There is no cart and no horse required.

Here's another way of looking at it: The only way we can attach meaning to the word "mathematics" is by its definition - it's rules - it's abstract framework. So while asking "How does math correspond to reality?" is an EXCELLENT question - science progresses on these questions all the time. BUT, the statement "Math is invalid or wrong or worthless unless it corresponds to reality." Is very different. It's either a personal value judgement, or a nonsensical statement/question, because it's definition, its nature, its framework if you will is separate from reality.

Bob

[Dragonfly]

Next, your definition of logic is one possible definition, but is not the most useful definition. The reason that mathematical reasoning, for example, appears useful is because it deals with numbers (among other things) and we know that numbers of things exist, e.g., two cows, three oranges, six billion people (give or take a few).

Groan... How many is i cows? Now Rand accepted the validity of the concept of imaginary numbers when she was told that they could be used for practical purposes (for example in electrical circuits). But that is such a narrow-minded idea that I feel I have to scream. Except for the subdiscipline of applied mathematics, the mathematician is not directly interested in the applicability of his theories (what is for example the practical aspect of the proof of Fermat's last theorem?). But what happens again and again is that purely abstract mathematical theories turn out to be very useful after all, but often many years or decades after they were developed (there are countless examples of this). Would they be "meaningless" in the meantime? Pure mathematics is a perfectly sound field which doesn't have to be validated by its practical applicability, which doesn't mean that it often turns out to be very practical after all. It is like Rand's defense of capitalism: she doesn't justify capitalism by its practical results, but that doesn't mean that capitalism isn't an efficient system.

But, imagine what would happen if we started with the symbols, "!@#$%^&*()_-+=~," made up some axioms governing them (e.g., if @ and #, then $) and then began to reason about them. It would be possible, but the excercise would be meaningless.

You should see some advanced mathematical texts...

[Dragonfly]

If you allow that a physical particle such as an electron could be (as it is presently thought to be in physics) a true single point having zero extent, then surely it is possible for there to be points of physical space that are true points having zero extent. And if such extensionless points of physical space are not to be excluded from physics, why exclude the breadthless lines that can be traced by such a point or the depthless surfaces that can be traced with such a line? Couldn't these be among the very lines and sufaces traced in space by moving bodies?

The question is whether "space" and "lines" have any meaning below the Planck limit.

[Rodney]

How many is -1 cows?

I don't really want to say much more on these topics at this point, except that I agree mathematicians often put forth consistent systems of entities and relationships as hypotheticals, and work out the implications without any thought of applications. And that these systems often prove to be useful and reflective of some truth.

[Michael Stuart Kelly]

I have a very basic question and I admit to being inclined to one side already.

Why can't mathematical reasoning be an innate ability that grows automatically on a basic level like language skills or logic?

Here is my reasoning from an Objectivist slant. Regardless of how "tabula rasa" a mind is at birth, it comes with the ability to integrate. That is pre-wired into the individual simply by being a member of the human species. Having the ability to integrate means having the ability to form units.

Also, Rand mentioned in her discussion of abstracting from abstractions in ITOE that man regards basic concepts as mental entities and starts integrating them to form higher concepts. This to me points to another innate capacity: the ability to regard a mental event as an entity and as a unit.

So if man has these innate capacities that develop independently of volition as he grows from infancy, why should the ability to form an abstract unit in his mind be excluded? The ability of being able to develop logic is innate, and this includes the ability to discern contradictory and noncontradictory on a mental level, not just a sensory level. (Interestingly, man has no choice about having volition, either. His ability to choose is innate and he has no choice at all about it. He can only choose how to exercise it.)

So I see no reason to try to force pure mathematics into the "derived from sensory data" mold. This is like saying that the ability to integrate comes from sensory data. The ability to integrate is already there. It came before the sensory data. It came at birth as part of man's nature. Then when the sensory data came, there was a built-in method for what to do with it.

It makes sense to me that if man has the innate mental capacity to form units and the innate capacity to form abstract mental entities, then he has the innate capacity to form abstract mental units as well—and these mental units could have the same fundamental axiomatic properties (especially identity) as any entity in external reality.

Michael

[bmacwilliam]

Dragonfly:

"But that is such a narrow-minded idea that I feel I have to scream."

Well, at least I'm not the only one.

And Michael, I tend to agree with what you've written:

"So I see no reason to try to force pure mathematics into the "derived from sensory data" mold."

Yes, but in my opinion, the problem is deeper than that.

Incidentally, I read a book last summer called 'How the mind Works' by Stephen Pinker. Interesting stuff along the lines of what you're talking about - all about learning and human nature/evolution etc. Very thought-provoking.

Bob

[Darrell Hougen]

I'm getting so many responses to my posts, that I can't respond to them all individually. However, let me try to clarify things a bit.

There are numerous features of reality that are (perhaps blindly) incorporated into the logic that we use, which is fine until we run up against cases in which it turns out that the features upon which our reasoning was dependent are just approximations. In predicate logic, there is a subject, which may be universally quantified, and a predicate that is taken to be true of that subject. Now, if we look closely, there are several features of reality that are reflected in predicate and mathematical logic

For example, there are individual entities (or actions), each with an identity. If reality was an amorphous mass that could not be divided into individual entities, the notion of a subject would be meaningless.

There are classes of objects that may be grouped together. For example, all oranges are members of the class of oranges. If there were no similarities between objects that existed, then no classifications could be formed. And, if no classifications could be formed, it would be useless to count things because there would never be more than one of anything and quantification would be virtually useless as well.

Space has a metric character in that measurements of distances are ordered and may be non-integral. Those notions are incorporated into our real number system.

Each of the above examples illustrates the manner in which some aspect of reality has been incorporated into our mathematical logic. The rules, and not just the symbols governing the relationships between mathmatical quantities, are dependent upon the nature of reality. The only universal element of logic that can be said to be independent of reality is the notion that no contradictions can be allowed into any logical system. But, even that is an admission of the primacy of existence as no contradictions can exist in reality.

When a new feature of reality is discovered that cannot be adequately explained using our existing notions of math and logic, new mathematical axioms must be added to our systems. Now, once we have a set of axioms, there is nothing inherently wrong or invalid about studying the system itself and attempting to determine its properties, irrespective of any application, present or future. The system itself is an interesting object of study. But that does not imply that the system is somehow divorced from its roots in existence. Notice, for example, that the axioms of integer arithmetic are not sufficient to explain the real number system. Nor is any finite set of axioms sufficient to predict all mathematical truths.

Notice also, that mathematical logic is an abstraction of what goes on with respect to real objects. For example, the statement, "All men are mortal," may have to be modified in the next few years or decades or centuries if the aging problem is solved. It will still be true that men may die of gun shot wounds or car accidents, but they will no longer die of old age. Similarly, any other concept of existing things may have exceptions or may have to be interpreted within a specific, limited context.

Darrell

[Guyau]

Bob, you wrote in #106:

"This is not to say tht logic can't predict anthing, and if a prediction turns out to be false, then, sure, maybe the logic is wrong, but logic is separate from evidence. I remember reading stuff on this by Putnam(?) I think . . ."

Yes, Putnam is a correct file in memory you were searching, and so is Quine.

It was Quine's essay "Two Dogmas of Empiricism," published in 1951, that brought his debate with Carnap over the analytic-synthetic distinction to widespread attention among philosophers. In this essay, Quine argued against the validity of the distinction. Carnap wanted to maintain a sharp distinction between analytic statements depending entirely on the meanings being used and synthetic statements making assertions about the empirical world. Quine's alternative view had it that all statements face the world as part of a corporate body of statements. On this view, experience bears the same kind of evidential relation to the theoretical parts of natural science as it does to mathematics and logic. (See also Quine's 1960 essay "Carnap and Logical Truth," which is in the collection The Ways of Paradox and Other Essays. "Two Dogmas" is in the collection From a Logical Point of View.)

During the 1950s, Putnam was also writing about the analyticity of various statements, such as the statement of Rand's in 1957 that a leaf "cannot be all red and all green at the same time." Other philosophers, too, such as Arthur Pap and Morton White, were writing on the analytic-synthetic controversy during the 50s.

In her journal The Objectivist, immediately after the the issues of the journal containing her "Introduction to Objectivist Epistemology," Rand published Peikoff's "The Analytic-Synthetic Dichotomy" (1967). His voice (speaking also for Rand) joined the voices arguing against the validity of the distinction.

Returning to Putnam, in 1968 he published the essay "Is Logic Empirical?" which was renamed "The Logic of Quantum Mechanics" in a later collection of his papers. He proposed a quantum-logical interpretation of quantum mechanics to give an example of how we might have an empirical reason for revising logic.

Reichenbach had attempted an interpretation of quantum mechanics in 1944 using a three-valued truth-functional logic. (Our standard logic is a two-valued truth-functional logic.) The Reichenbach quantum-logical interpretation of quantum mechanics has no adherents today.

"Far more important than artificial many-valued logics are the quantum logics that arise naturally in the Hilbert space formalism. The strongest of these logics---strongest in the sense of most closely approximating to classical logic---is the logic which mirrors the structure of the set of closed subspaces of Hilbert space. This quantum logic was first studied by the mathematicians Garrett Birkoff and John von Neumann in a classic paper of 1936. By quantum logic we mean Birkoff and von Neumann's quantum logic and the various ways in which it may be formulated as a logic.

"A lattice of propositions is of course not quite a logic, whatever a logic is. But a lattice of propositions has a structure which is at least very much like the structure of a logic. So if quantum logic really is a logic, we should first make it look like a logic. Then we will be in a position to discuss whether it really is a logic.

//

"Quantum logic can in fact be fairly easily transcribed as a logic in the usual logical styles---as an axiomatic system, as a sequent calculus, and as a natural deduction system. . ." (127-28).

Particles and Paradoxes: The Limits of Quantum Logic

Peter Gibbins (1987)

Quantum logics utilize algebraic accounts of quantum theory. They make use of Boolean algebras, partial Boolean algebras, and orthomodular lattices. These structures can be found embedded in Hilbert spaces, those complex topological vector spaces appropriate to quantum mechanics. [see Chapter 7 of The Structure and Interpretation of Quantum Mechanics R.I.G. Hughes (1989).]

The basic idea is to supercede the classical logical constants of AND, OR, and NOT with the lattice-theoretic (algebraic) operations of MEET, JOIN, and ORTHOCOMPLEMENT. Recall the distribution equivalences for AND and OR that we learn in elementary logic, such as {[p AND (q OR r)] Equiv [(p AND q) OR (p AND r)]}. Corresponding equivalences of distributions do not obtain for MEET and JOIN in the algebra appropriate to quantum mechanics.

It seems to me as to most philosophers of physics today that the word supercedes in the first sentence of the preceding paragraph gets it wrong. The very Hilbert spaces and algebraic structures embedded in them from which some would draw a new logic are themselves deductively certified as sound mathematics using simply classical logic.

Edited February 23, 2007 by Stephen Boydstun

[bmacwilliam]

Thanks for the info Stephen - much appreciated. There's enough reading there to keep me busy for the next few months in my non-existent (but logically valid) spare time :-)

Bob

[Michael Stuart Kelly]

Bob,

LOL...

Dayaamm!

You're growing a sense of humor.

Must be that unpredictable quantum stuff acting up in your brain.

Michael

[Darrell Hougen]

Lol. I can't follow jokes like those. Thanks, Stephen, for the reading suggestions.

Darrell

[Paul Mawdsley]

Darrell,

I'm still learning the formal language to talk about these subjects. I will take your lead and refer to the distinction between the two modes of reason as "predicate logic" and "mathematical logic."

I maintain that there is a more developmentally (and evolutionarily) basic form of logic that operates at an intuitive level of reasoning whereby we are able to use epistemological principles of identity and causality to manipulate experiential images directly. It is from such processes that our intuitive understanding of events and our intuitive philosophy evolve. Linguistic and mathematical reasoning are special abstract cases of this more developmentally basic type of reasoning. These two formal symbolic forms of logic apply a special case of symbolic identity and causality to create statements and arguments.

Linguistic and mathematical reasoning are rooted in two realms---i.e.: the realms of words and numbers---that were created by the human mind, but can be used to represent elements of reality and manipulated to help integrate information about reality. We are able to use linguistics and mathematics as programming languages to interface with and shape the content of our imagination and/or intuition. This brings our worldview within our conscious and volitional control. Unfortunately, it can also split our intuitive experiential perspective and our conscious symbolic perspective into competing worldviews, which often leads us to a polemic between conscious and subconscious perspectives in which we ultimately chose one over the other and become disconnected from part of ourselves.

I have suggested that we should be able to apply our imaginations, in the spirit of philosophy, for the purpose of understanding our existence and generating our worldview. In the context of the current thread, I suggest we should be able to explore the possible underlying structures and dynamics of quantum events. This suggestion only makes sense if we are to use well defined ontological concepts of identity and causality as guiding epistemological principles. These principles must account for and integrate all observations and evidence. They must be abstracted from and tested against observation and evidence.

This is what I mean by an empirically expanded version of identity and causality. As metaphysical axioms, identity and causality have limited practical usefulness. As ontological and epistemological principles, informed by observation and evidence (including evidence from linguistic and mathematical perspectives), identity and causality can be used as guides to building models of existence using experiential/intuitive images.

You have mentioned the Michelson-Morley experiment. This, combined with Special Relativity and quantum mechanics, has caused modern physics to divorce itself from the enterprise of building experiential/intuitive models of existence that adhered to our notions of identity and causality. The ether theory was the last such theory, and it was clearly demonstrated to be mistaken. Now it is said that our common-sense notions of identity and causality cannot create an experiential/intuitive model of existence that can fit the evidence and our mathematical descriptions of the universe, and so we must dispense with common sense as a means to knowledge. I agree that our common-sense notions of identity and causality are insufficient to understand existence. They are ontologically weak. I disagree that this means we should dispense with common-sense as a means to knowledge.

We need to change our notions of identity and causality so they fit existence. Ayn Rand and Albert Einstein each demonstrated the use of different but incomplete notions of identity and causality. What if their views were integrated, our common-sense notions were raised to a higher level, and we tried again to integrate the evidence into a "common-sense" worldview? What would quantum reality look like from an integrated Einstein/Rand perspective of identity and causality? In essence, this is what I have been playing with for some time. I think it can lead to an intuitive or common-sense understanding of existence that integrates and interprets the evidence from a more solid ontological foundation than wave/particle dualism, virtual particles and the collapse of causation into probabilities.

Paul

Edit: Changed the last word from randomness to probabilities. It strikes me as being more precise.

Edited February 24, 2007 by Paul Mawdsley