Your optional course selection is pretty similar to mine, actually -- I'm taking the Biddle, Thompson, Brook and Lewis courses, as well as the Harriman and Rheins. I guess we have common intellectual interests.

Will there be a Noodlefood BOF at the con? Seems like it might be a good chance to put names to faces. 1173911099:::PMB:::::::::Those are all great choices, Diana. But you left off the course I'm most excited about (besides Dr. Peikoff's): Alex Epstein's course, "The Media's Fraudulent Accounting of Business Scandals." He's an excellent speaker, and if his press releases on Enron are any indication, his presentation is going to blow away everything people think they know about that so-called "scandal." 1173911898:::Diana Hsieh:::diana@dianahsieh.com:::http://www.dianahsieh.com/blog:::PMB: I've not seen Alex speak before, although he did teach one of my "Intermediate Writing" classes -- just one class, not the whole course -- and it was excellent. However, I just cannot muster any enthusiasm whatsoever for the topic of the recent business scandals. If I hear great things about the course, I'll order it on CD though. If it's as good as you suggest it might be, then it'll be well worth hearing.

Admittedly, it would be a topic worth my knowing about, particularly if I ever teach politics, as I surely will someday. (I don't right now, although goodness only know what I'll be teaching next semester! I've decided that I'll teach anything -- even German phenomenology -- so long as I don't have an 8 am class.)

Kyle: I'll be delighted to see you at OCON. But what's a BOF? 1173921951:::John Stark:::raistlan@atheist.com:::http://starkrelief.blogspot.com:::OCON is the main part of our honeymoon, so Arwen and I will be there. 1173923586:::Matt F. :::deadly_vu_666@yahoo.com::::::I'll be there. I'm an OCON virgin (hey, I'm young). I desperately wanted to attend last year--scheduling conflict. Peikoff's lectures and the fact that I so dearly miss Colorado had me determined to make it out this year.

Off the top of my head, I do not recall what groups I am in for each particular course, but the courses I am taking are:

ECONOMICS Part 1 by B. Simpson -- Although I can easily shoot down most of the nonsense that my peers throw (or, more precisely, awkwardly fumble towards) at me, I still want a more fundamental grasp of the subject as a whole.

ATLAS SHRUGGED AS A WORK OF PHILOSOPHY by G. Salmieri

TWO, THREE, FOUR AND ALL THAT by P. Corvini

PHILOSOPHY OF IMMANUEL KANT by J. Rheins

I'd like to take more courses but, well, I'm a college student and therefore poor. My interests in philosophy lie predominantly in metaphysics and epistemology so I my selections are based largely on that. I had a hell of a time crossing courses off the list that I wanted to take but couldn't afford to.

Hope to see you all there. I'll be the one in the dark suits with long black hair.

Matt F. 1173950205:::Kyle Haight:::khaight@alumni.ucsd.edu:::http://www.leftist.org/haightspeech/:::Diana,

"BOF" is a term I picked up from technical conferences; I think it may also be used in science-fiction fandom. It stands for "Birds Of a Feather". An "X BOF" is a gathering of people interested in or associated with X. In this case a Noodlefood BOF would be a gathering of Noodlefood contributors, commenters and readers. (At technical conferences they sometimes set aside rooms that can be used for this sort of thing, and BOFs will be spontaneously organized by conference attendees. In the OCON case, it might be more appropriate to pick a day and go to lunch as a group or something like that.) 1173981034:::Monica:::monicabeth10@gmail.com:::http://sparkasynapse@blogspot.com:::I'll be attending for the first time. Very much looking forward to it! 1173984840:::Mike Hardy:::hardy@math.umn.edu::::::Searching for Pat Corvini via Google Scholar I find

various writings on engineering and none on philosophical

foundations of mathematics. On regular Google I find

Stephen Speicher saying he was disappointed in a talk

of hers on the nature of infinity. He says she called

it a concept of method and treated it be conventional

epsilon-delta methods. My generally dormant cynical

side makes me wonder if the only "infinity" treated is

the one that occurs in calculus. What about the "infinity"

Euclid dealt with in his famous proof that infinitely

many prime numbers exist? Certainly that's more basic

than the one encountered in calculus and treated by

"epsilon-delta" methods. So far she's keeping her

cards a bit close to the chest by publicizing the idea

only in a talk at this sort of meeting, so such hearsay

as Speicher's comments may perhaps mislead and have

nothing authoritative to contradict it. So if anyone

knows anything, pleasantly surprise me if you can by

refuting my cynical suspicions. 1173995870:::Burgess Laughlin::::::http://www.aristotleadventure.com:::In Comment ID: #8, Mike Hardy says: My generally dormant cynical side makes me wonder [....] So if anyone

knows anything, pleasantly surprise me if you can by refuting my cynical suspicions."

The term "cynicism," for me, names this idea: The belief that virtue is impossible, and so we must look for other, hidden and base motives behind the actions people take.

Is that what you mean by "cynicism"? 1173996212:::Diana Hsieh:::diana@dianahsieh.com:::http://www.dianahsieh.com/blog:::Mike -- As far as I recall, Pat Corvini is working on a book at present. (Until recently, she worked in industry, not in academia. So it's not surprising that she hasn't published.) Her course was excellent, in my judgment. (And I wasn't alone.) That Stephen Speicher thought ill of it... well, that's just one more reason to think ill of him.

As for what the course actually says, you should buy or borrow it, so that you can hear it for yourself. Or you can wait for the book. I'm not going to summarize it here, nor entertain further discussion of it based on SS's comments. That wouldn't be fair to Pat. 1173997166:::Diana Hsieh:::diana@dianahsieh.com:::http://www.dianahsieh.com/blog:::In particular response to Mike's comment -- "So if anyone knows anything, pleasantly surprise me if you can by refuting my cynical suspicions" -- I would like to add that the wholly unjustified cynicism of his suspicions is the primary reason why I have no interest in or even tolerance for his inquiries. Pat deserves better, particularly from someone without any first-hand knowledge of her work.

Mike, haven't we already gone around on this issue on more than one occasion with respect to PARC?!? There's a principle involved here, a principle of justice. Until you actually respect it -- at least in my comments -- I'm going to continue to be annoyed by your arbitrary "cynical suspicions." 1174047998:::Mike Hardy:::hardy@math.umn.edu::::::Burgess Laughlin wrote:

:: My generally dormant cynical side makes me

:: wonder [....] So if anyone knows anything,

:: pleasantly surprise me if you can by refuting

:: my cynical suspicions."

::

:: The term "cynicism," for me, names this idea:

:: The belief that virtue is impossible, and so

:: we must look for other, hidden and base motives

:: behind the actions people take.

::

:: Is that what you mean by "cynicism"?

Nope. Rhetorical license. Obviously.

Diana wrote:

:: would like to add that the wholly unjustified

:: cynicism of his suspicions is the primary reason

:: why I have no interest in or even tolerance for

:: his inquiries. Pat deserves better, particularly

:: from someone without any first-hand knowledge of

:: her work.

My suspicions arise from various earlier claims by

various people to have settled questions in the

philosophical foundations of mathematics via

Objectivist epistemology that have failed to pan out.

:: Mike, haven't we already gone around on this issue

:: on more than one occasion with respect to PARC?!?

No. How is this related to that? 1174049788:::Mike Hardy:::hardy@math.umn.edu::::::If I were to write "Based on some previous

experiences I am generally very pessimistic

about claims to have settled questions in the

philosophical foundations of mathematics by

using the Objectivisit epistemology", I wonder

if it would be inferred not only that I meant

"pessimistic" in one of the technical senses

in which philosophers use that term but also

that that is my attitude toward the universe

in general? 1174066901:::Burgess Laughlin::::::http://www.aristotleadventure.com:::Mike, Comment 12:

:: Is that what you mean by "cynicism"?

"Nope. Rhetorical license. Obviously."

Obvious to whom? Not to me. Hence my question. By the way, what *did* you mean by "cynicism" -- or, below, by "pessimism" -- that couldn't be conveyed by a word such as a form of "doubtful"? I am especially intrigued because you said this "side" of you is generally dormant.

Mike, Comment 13: "If I were to write 'Based on some previous experiences I am generally very pessimistic about claims to have settled questions in the philosophical foundations of mathematics by using the Objectivisit epistemology', I wonder if it would be inferred not only that I meant

"pessimistic" in one of the technical senses in which philosophers use that term but also that that is my attitude toward the universe in general?"

You use the passive voice, "be inferred." Inferred by whom? In what context?

Depending on the answer to the second question, and perhaps others, yes, I might infer it, but I can't speak for others.

One of the problems with internet communications, in my experience, is lack of sufficient context for particular comments. As Dr. Peikoff indicated recently on his website, in the Q&A section, questions in philosophy (and, I would suggest, most other subjects) cannot be answered with a simple yes or no. Context is crucial. Asking questions is one way of building some context. 1174070695:::Mike Hardy:::hardy@math.umn.edu::::::Sigh.

I'd have stuck to prosaic language if I'd

suspected I'd get construed so literally.

When Ayn Rand said she was interested in

"philosophy for living on earth" I thought

"living on earth" was metaphor expressing

exclusion of belief in the supernatural,

and I never expected to be writing patient

explanations that it does not mean "as opposed

to living on Mars, Venus, etc."

:: Obvious to whom?

To anyone I thought might read this page.

:: what *did* you mean by "cynicism" -- or,

:: below, by "pessimism" -- that couldn't be

:: conveyed by a word such as a form of "doubtful"?

I meant I don't like suspecting that the work

in question will not be fruitful. But previous

experience requires it.

One of the reasons I was surprised by Diana's

comparison to my comments about PARC is that

that is _not_ anything remotely resembling

cynicism even in this metaphorical sense.

Prosecuting a case holding that certain writers

lied is quite a different thing from contributing

to the philosophical foundations of mathematics.

In the latter case, I am very pleased if the

project succeeds.

:: Inferred by whom? In what context?

In THIS context, by those who post here. 1174075451:::kishnevi:::jbennetsmith@hotmail.com::::::I see no objectionable cynicism here from anybody, even in the most metaphorical sense.

But I think that presenting a satisfactory philosophy of mathematics is not possible from an Objectivist standpoint, because I don't see a way of doing it without falling back on Kant and his a prioris. And we all know what Objectivism's attitude to Kant is :)

I say that because a truly satisfactory philosophy of mathematics has to answer a question that does not arrive in any other theory of inquiry. Why does the universe appear to operate in a way that is intimately linked to mathematical principles. Kepler's laws are mathematical formulae; so is E=M*Csquared. [I'm not geeky enough to type superscripts.] Most science is descriptive; even when it deals with math, it's usually applying statistics, probability, calculus or geometry to physical data. Physics--the discipline that most nearly approaches the question of what the universe is like--does not. [This observation also applies in part to Chemistry, but only in part, and mostly only to that part where chemistry merges with physics.] It expresses itself through mathematics. Why is that so? Why is not merely possible, but necessary, for physics to express itself through mathematics? Why is there a precise mathematical relationship between energy and mass, and between the speed of a planet and its orbit around the sun? 1174084819:::Andrew Dalton::::::http://witchdoctorrepellent.blogspot.com:::kishnevi: Mathematics was developed by human beings for many purposes, precisely because it is useful. There is no reason to take that usefulness as a license to infer a world of perfect forms. That unjustified leap is similar to what creationists do when they are amazed that living organisms are equipped so well to survive on Earth, and then infer a supernatural Designer.

As far as why there is a precise mathematical relationship between this and that, the answer is simple: the law of identity. To be is to be *something*; and in many (but not all) cases, mathematics is the best conceptual tool for expressing what that something is. 1174143988:::Mike Hardy:::hardy@math.umn.edu::::::

:: Mathematics was developed by human beings

:: for many purposes,

True.

:: precisely because it is useful.

Really? How is Euclid's proof of the infinitude

of primes useful? How is the Riemann hypothesis

useful? It is true that methods of adding and

multiplying numbers, etc., may have been developed

initially because of usefulness, and the same is

true of some things in geometry. But it very often

happens---perhaps even more often than not---that

mathematicians introduce concepts and prove theorems

that are then found only decades or sometimes

centuries later to be useful. Only a couple of

decades ago algebraic geometry was a highly respected

research area in mathematics that most people thought

of as lacking scientific applications. Its

respectability came from other things than applications.

Now it is being applied to molecular biology and

other things, and people who learned it in 1980 or so

have expressed surprise at seeing it applied. Things

like that have frequently happened in the history of

mathematics. Yet here you say "precisely because it

is useful" it was developed. To defend that thesis

is going to cost you a lot of work. 1174149736:::A(ndrew) West:::andrew.h.west@gmail.com::::::I'll be there but am not taking many optionals. Staying at my former boss' house on the other side of the golf course from the conference location. I'm taking Y. Brook's and G. Salmieri's courses in session 2. 1174152711:::Andrew Dalton::::::http://witchdoctorrepellent.blogspot.com:::Mike: All valid concepts ultimately rest upon perceptual data, or upon a chain of higher-level concepts that themselves can ultimately be reduced to the perceptual level. Any logically sound connections, generalizations, and even hypothetical extensions (e.g., complex numbers) of these concepts have potential value--even if the application of these high-level integrations to the outside world (e.g., physics, engineering, etc.) is not immediately evident. Given the power that concepts give us over perceptual-level animals, we should be surprised if such connections did *not* exist.

The people whom it is "going to cost a lot of work" are those upon whom the burden of proof properly rests--that is, those who defend the idea of a parallel reality as the source of mathematics. 1174166693:::kishnevi:::jbennetsmith@hotmail.com::::::Andrew, I think you misunderstand me. I am saying that there exists a certain relationship between mathematics and the makeup of the physical world. How do you explain this WITHOUT resorting to a world of forms, or a prioris, or the Mind of God?

The law of identity is not sufficient. That's no more than saying "that's the way it is". WHY is it the way it is? Why is it that, for physics, "mathematics is the best conceptual tool for expressing what that something is"? Why is the relationship among mass, energy, and the speed of light capable of being expressed in mathematical terms, when mathematics is superficially a purely mental construct?

I don't think that Objectivism can provide a satisfactory answer to this, which means that it can't provide a satisfactory philosophy of mathematics. If someone can do it, my hat is (prospectively) off to them. 1174223788:::Freddy Ben-Zeev:::benzeev(at)comcast(dot)net::::::kishnevi wrote: The law of identity is not sufficient. That's no more than saying "that's the way it is".

Saying "that's the way it is" is perfectly valid explanation when you refer to the primary ideas in a given context. For the law of identity, the context is the whole reality. It means "this is what we observe in reality". What would be the alternative? Going outside of reality?

kishnevi wrote: ... mathematics is superficially a purely mental construct.

Mathematics is not a purely mental construct. Like all knowledge, it is build on concepts abstracted from reality (e.g. relations of count, size, etc), using methods derived from reality (i.e. logic). In short it is neither subjective, nor intrinsic, but objective. To be surprised at the way mathematics fit reality, is no different than being surprised at how concepts fit reality. 1174230159:::Mike Hardy:::hardy@math.umn.edu::::::Andrew Dalton wrote:

:: The people whom it is "going to cost a lot of

:: work" are those upon whom the burden of proof

:: properly rests--that is, those who defend the

:: idea of a parallel reality as the source of

:: mathematics.

Why is such an idea as "a parallel reality as the

source of mathematics" being brough into this

discussion? Does someone take that position?

It may well be true that all "All valid concepts

ultimately rest upon perceptual data" and that

"the application of these high-level integrations

to the outside world (e.g., physics, engineering,

etc.)" may always be there even if they are "not

immediately evident." But originally you said

that such things were developed precisely because

they are useful. That is false, and seems to

conflict with your suggestion that they may be

developed even when their application is not

immediately evident. 1174236146:::Fred Weiss:::fredweiss@papertig.com:::http://www.papertig.com:::"kishnevi" asks why the relationship of physical phenomena (such as mass, energy, and the speed of light) can be expressed in mathematical terms if mathematics is "a purely mental construct".

I don't know who here is claiming that it is "a purely mental construct" or if that is just "kishnevi's " personal view. It is certainly not the Objectivist view.

The reason why physical phenomena can be expressed in mathematical terms should be obvious. They possess properties and attributes and perform actions which are measurable. For example, they have size, weight, shape or speed. They bear certain measurable characteristics in relation to each other, such as distance, gravitational pull, etc.

Why should these measurements require "resorting to a world of forms, or a prioris, or the Mind of God"?

The fact that specific measurements, qua measurements, don't exist, i.e. are not separate existents by themselves doesn't mean therefore that they are solely figments of our imagination. True, "one inch" doesn't exist. You can scour the universe from one end to the other and never find "an inch". But that two objects are "an inch apart" is a fact of reality which we can measure. It is a unit of measurement in a method we have devised to measure distance - and distance is a fact of reality which exists between entities. 1174239526:::Fred Weiss:::fredweiss@papertig.com:::http://www.papertig.com:::Mike Hardy raises a much more interesting question regarding mathematics.

He wants to know what is the relationship to reality of certain very advanced mathematical concepts both in their original conception and in their eventual application. That's *the big* question in my view in the philosophy of mathematics.

Mathematicians clearly pursue their work with no apparent connection to reality - that is other than previously known or more basic mathematics upon which it is based. It appears to be an edifice in something of its own reality, the validity of which derives solely from logic and principles of mathematics which may be entirely of the mathematicians creation.

Mike however makes a very suggestive comment which I have heard many times before from knowledgeable mathematicians/physicists. He tells us, "mathematicians introduce concepts and prove theorems that are then found only decades or sometimes

centuries later to be useful." And he offers the following example, "Only a couple of decades ago algebraic geometry was a highly respected research area in mathematics that most people thought of as lacking scientific applications...Now it is being applied to molecular biology and other things, and people who learned it in 1980 or so have expressed surprise at seeing it applied."

It is my understanding that the same thing can be said about virtually all advanced mathematics. Note how calculus - not known previously - was used by Newton to express his brilliant and groundbreaking work in physics.

I have an hypothesis. It's very speculative and I tread lightly with it because I know almost nothing of either advanced mathematics or physics. So, with that proviso, take this for what it may be worth as a potential kernel of thought.

I don't think it's a coincedence that almost all mathematics, however advanced and obscure eventually seems to find applications in science. The reason I think is that the mathematics itself drives the science! Since science requires mathematics, the very existence of some new mathematics will make some science possible that otherwise might not have been possible - or not possible to any advanced degree.

But the question is why that will seemingly happen to even the most abstract and obscure mathematics, where the mathematicians cannot imagine any applications. I think it's because the degree of complexity of the universe is huge, possibly even limitless. It is possible therefore to look at it from innumerable different perspectives which mathematics makes possible and where that particular perspective may not have been pursued except for the existence of the mathematics.

As I said, this is just an hypothesis and not one I can do much with without knowing much more mathematics/physics than I do. So, it will have to be for someone else with such knowledge to flesh it out if they think it has any explanatory value. 1174248817:::Andrew Dalton::::::http://witchdoctorrepellent.blogspot.com:::Mike-

The issue of a "parallel reality" was being defended by kishnevi; I'm sorry if I incorrectly implied that you were taking that particular position.

On the question of abstract mathematics being far ahead of applications:

I think that Fred Weiss is on the right track regarding the enormous complexity of the universe, although I am not ready to accept (entirely) the idea that abstract mathematics is driving its own applications in science. As I will explain later, this approach has been done and can lead to wishful thinking taking the place of real understanding.

My own tentative view is simply that the complexity of the universe almost guarantees that mathematics will be useful somewhere, in some context. The consistency of reality (the law of identity) and the consistency of mathematics ensure that these correspondences will occur much more frequently than by chance. (This is what I meant by mathematics being "useful" aside from its obvious descriptive applications.) However, exactly *where* a particular area of mathematics will be applicable is not something that can be predicted without observations.

I should stress this last point--mathematical reasoning alone does a lousy job of predicting its own applications. One example is the Golden Ratio (phi), which has been known since ancient times. This number has been applied by mathematicians and philosophers to countless ratios in art, nature, the human body, etc. A great deal, if not most, of these correspondences are accidental. A few--notably the spiral growth of plants and the structure of quasi-periodic crystals--are empirically and theoretically ironclad. In these latter cases, it is the union of abstract and the empirical that makes the mathematics truly useful for understanding the universe. 1174261540:::Fred Weiss:::fredweiss@papertig.com:::http://www.papertig.com:::Andrew, by "driving its own applications in science", I don't mean to suggest that every application of mathematics is necessarily valid. Some of it could even be entirely arbitrary. I would be extremely suspicious of the mathematics that is used to justify "string theory" for example.

However you have drawn attention to the aspect of my hypothesis of which I am most unsure myself. But offhand I don't know how else to explain how even the most seemingly bizarre and extremely abstract mathematics eventually seems to find scientific and/or technological applications. The alternative is that when scientists are exploring some new phenomenon which they are trying to understand a light bulb goes off and someone remembers some obscure mathematics that could explain it.

I really don't know but I'd be interested in hearing from those who have actual knowledge and experience with this subject.

But I'm more drawn to my hypothesis concerning the "connection to reality" of mathematics - even very, very abstract - possibly even seemingly bizarre - mathematics. I think that the number of ways that the different aspects of the universe and their relationships could be expressed mathematically is virtually infinite. So that no one could possibly ever dream up all the possible mathematics to explain it.

Mathematicians can never work themselves out of a job. More "revenge of the nerds". :-) 1174332223:::michael:::michael@m(eye)rmak.com::::::Back on topic, I'll be at OCON for every single minute. 1174337433:::kishnevi:::jbennetsmith@hotmail.com::::::In response to Mr. Dalton:

To be very clear (and I thought I was being clear at the outset), I am not defending the "parallel reality" explanation. (Although I would not call it "parallel"; but that is another discussion entirely.)

My observation was simply this (and I am putting it as simply as possible so it is as clear as possible): since Objectivism rejects "parallel reality" on principle, how can it explain a certain characteristic of mathematics which I happen to believe is the most important question which can be considered by a philosophy of mathematics. I expressed my belief that because of its fundamental principles, no philosophy of mathematics based on Objectivist principles can answer that particular question--but I also stated that I could very well be wrong about that. I referred to "parallel reality" only to note that, while from an Objectivist standpoint it is inadmissable, a satisfactory answer to my question may not be possible without it. I would not call that a defense. You would think I had brought the Sefer Yetsirah into the conversation: "Ten ineffable Sefirot--ten, not nine; ten, not eleven" But I didn't. Please don't say I did what I did not do.

One person raised the Law of Identity. I responded that was no real answer to the question--merely a statement that the particular characteristic existed.

Mr. Weiss then proposed what seems to me to be a descriptivist view of mathematics, and then in his next comment rephrased my question in a way that apparently is more coherent and therefore more understandable to the readers here than my original formulation--and I thank him for that.

To Mr. Weiss, beyond the thank you:

Is your distrust of the mathematics associated with string theory due to a distrust of string theory, or to a distrust of the math itself. (I have a general knowledge of string theory, but not with the math involved.)

And, to refresh your memory, calculus was not merely unknown until Newton's time. He invented it for the sake of his physics (with of course some competition from Leibniz.) 1174386529:::Fred Weiss:::fredweiss@papertig.com:::http://www.papertig.com:::"kishnevi" asked me about my distrust of string theory.

It is more than just distrust.

String theory as I have heard it expressed, and as I understand it, rests on a stolen concept of "multiple dimensions". One has to ask multiple dimensions *of what*?

It can only be of one thing: reality.

But there cannot be another dimension...of reality. Is that other dimension real? Then it is part of reality, not "another dimension".

True, as reality exists there are some differences in relation to us. For example, some of it we can perceive with our unaided senses and some we can't. But that which we cannot see (with the unaided eye) is not in another dimension - nor could it be. We just need other mechanisms to discover its properties, e.g. microscopes.

There is not - nor could there be - some reality beyond our capacity to discover, some ineffable dimension(s) with mysterious, unknowable properties.

This is why, btw, I have heard it said by some that string theory is in principle unverifiable. Well, of course, how can we study some "dimension" which is beyond man's cognitive capacity? You have to ask what kinds of properties would be beyond our capacity to know? Well, you could say, it has such and such property which is beyond our grasp. Then how do you know it has that property and what it is?

The whole thing is an epistemological disaster and you don't have to be a physicist to see it. 1174409828:::Mike Hardy:::hardy@math.umn.edu:::::::: But there cannot be another dimension...of

:: reality. Is that other dimension real? Then

:: it is part of reality, not "another dimension".

This is utter nonsense. The meaning of the word

"dimension" being used in the quote above is

completely unrelated to the one used in physics

and mathematics.

Geometry in spaces of a large number of dimensions

is used in many areas of pure and applied mathematics.

Simple example: In statistics you are taught how to

find a confidence interval based on a small sample

from a normally distributed population. The

validity of the technique for finding such an

interval relies on the sample variance having a

chi-square distribution, with a number of degrees

of freedom equal to one less than the sample size?

How do you know it does? The slickest elementary

argument proving this applies more-or-less ordinary

Euclidean geometry, but in a space in which the

number of dimensions is the sample size---typically

much larger than the three that Euclid worked with.

Your great leap to a conclusion about what the word

"dimension" means in the context you refer to is silly

and childish. 1174412893:::Burgess Laughlin::::::http://www.aristotleadventure.com:::In Comment 31, Mike Hardy says: "The meaning of the word 'dimension' being used in the quote above is completely unrelated to the one used in physics and mathematics. [...]

Your great leap to a conclusion about what the word 'dimension' means in the context you refer to is silly and childish."

What is your serious and adult definition of "dimension" in physics and mathematics? (Is it the same in the two fields?) 1174414662:::Mike Hardy:::hardy@math.umn.edu:::::::: What is your serious and adult definition

:: of "dimension" in physics and mathematics?

:: (Is it the same in the two fields?)

There's a whole slew of very much related concepts

that go by that name, but let's start with a really

simple one: A line is 1-dimensional; a plane is

2-dimensional; the physical space we live in is

3-dimensional. You can specify a physical location

by specifying three numbers; think of longitude,

latitude, and altitude or the like. (This doesn't

seem like a good forum of any lenthier account of

this topic.) 1174414889:::Mike Hardy:::hardy@math.umn.edu::::::And Fred Weis: please refrain from making a total

crackpot of yourself.

Saturday Night Live's Emily Litella would wax

passionate in an editorial against the Eagle

Rights Amendment or measures to conserve our

natural racehorses or the like. Someone would

point out her misunderstanding and she uttered

her famous two-word phrase and withdrew.

Not a good exemplar. 1174418255:::kishnevi:::jbennetsmith@hotmail.com::::::Mr. Hardy has already pointed out that mathematics of n-dimensions is a valid part of mathematics. The relationships of those dimensions, when n > 4, to the physical dimensions of the universe, is, from what I know of the subject, a very undecided matter. The fifth dimension may be to time what time is to space, and what space is to area, and what area is to length--but the mathematics of five dimensions does not depend on that. The only people who have dealt with moving about in six physical dimensions are the Gallifreyans, and for obvious reasons, their experience is irrelevant to this discussion. But I find the following statement of Mr. Weiss to be a bit troubling:

There is not - nor could there be - some reality beyond our capacity to discover, some ineffable dimension(s) with mysterious, unknowable properties.

It is legitimate to say that we should only speak of, because we can only know, the physical universe as revealed through our senses. It is quite another thing to say that, as a matter of principle, existence is limited only human experience (actual or potential) and that it is impossible for anyting to exist which can not be experienced through human senses. In fact, the assertion bears a paradoxical aspect. To know as a certainty that something does not exist, we must be able to experience its absence--its non existence. But how can you experience the absence of something you can not experience in the first place? You can't know something is not there unless you can also (potentially) know that it is there. You can only state that you have no experiences which bear on the question, and therefore can make no rational statements about the something-that-may-or-may-not-be-there. 1174418468:::Burgess Laughlin::::::http://www.aristotleadventure.com:::Comment ID: #33 (link)

In Comment 33, Mike Hardy quotes me:

:: What is your serious and adult definition

:: of "dimension" in physics and mathematics?

:: (Is it the same in the two fields?)

And then Mike says: "There's a whole slew of very much related concepts that go by that name, but let's start with a really simple one: A line is 1-dimensional; a plane is 2-dimensional; the physical space we live in is 3-dimensional. You can specify a physical location by specifying three numbers; think of longitude, latitude, and altitude or the like."

Since you didn't answer either of my questions, I will repeat them:

1. What is your serious and adult definition of "dimension" in physics and mathematics?

2. Is the definition the same in the two fields?

Mike adds: "(This doesn't seem like a good forum of any len[g]thier account of this topic.)"

One way to shorten the discussion is to answer the questions -- or say that you can't. If the latter is the case, you might reconsider your approach to Fred's argument. 1174420843:::Fred Weiss:::fredweiss@papertig.com:::http://www.papertig.com:::"kishnevi" asks why "it is impossible for anything to exist which can not be experienced through human senses."

Suppose something did exist which was undetectable to our senses. Bear in mind this has to include not having any discernible effect on anything which we can perceive. There are many things which we cannot perceive (directly) but the existence of which we can deduce from its effects on things which we can perceive - such as a measuring device.

What properties would such an undetectable existent have? Undetectable ones. Well, what are those? I don't know, they are undetectable. So how do you know it exists? I don't. Actually, I can't by definition.

So what is the point of hypothesizing about such a thing? It will never make the slightest difference to us. (If it did that would be a discernable effect which we could perceive).

To respond to your seeming paradox, yes, you are right, I cannot prove that such a thing doesn't exist. But the burden of proof is not on me to prove that such a thing *doesn't* exist, which as you correctly point out is impossible. The burden is on he who hypothesizes it. But that also is impossible by definition.

Either way you look at it, "kishnevi", it's an impossibility.

P.S.: Not to overburden our good friend, Mike Hardy, who seems at the moment to be having difficulty getting past arm-waving and unsupported pronouncements, I would like, Burgess, if I may, to add a 3rd question to your list.

3. And is that sense of "dimension" the one that applies to string theory 1174421697:::Fred Weiss:::fredweiss@papertig.com:::http://www.papertig.com:::Incidentally, I hope we are not going to be subjected to a clearly equivocal sense of "dimension" which is something like "a different way of looking at something". For example, another dimension on this problem is the socio-economic or its effect on women or blue-eyed, red-haired children - or whatever, etc. etc. In this sense there is in fact only one thing, one existent, but it is being looked at from different perspectives, potentially many, even in some cases validly.

That is entirely different from my understanding of what is being proposed in string theory. 1174480077:::Mike Hardy:::hardy@math.umn.edu::::::I did not state a definition, but what

I did state should be quite informative

to anyone who didn't know that "dimension"

doesn't mean anything like the things

proposed here. Let's try a definition

valid in at least some of the relevant

contexts: the dimension of a vector

space is the maximum number of vectors

in the space that can be linearly independent,

and hence is the number of scalars needed

to specify any one particular vector.

For now, just think of "scalar" and "vector"

as meaning what you were taught in high

school.

More later.... 1174484549:::Mike Hardy:::hardy@math.umn.edu:::::::: One way to shorten the discussion is to

:: answer the questions -- or say that you

:: can't.

I can indeed answer this, and as you see I've

given part of an answer above. I was rushed.

See more below.

:: If the latter is the case, you might

:: reconsider your approach to Fred's

:: argument.

No reason to. He was just being abusive

by writing about what he was unwilling to

know about. Or maybe didn't suspect that

there was anything to know, before issuing

his bizarre condemnation of what he didn't

know about.

For present purposes, one can take the

word "scalar" to be synonymous with

"real number". (In some other contexts,

perhaps one could say especially in physics,

it means "complex number", i.e. a number

with a real part and an imaginary part.

Please don't take the words "real" and

"imaginary" literally; they are entirely

standard misnomers; given some of the

silliness above I feel I have to point

this out. And in other contexts "scalar"

can mean "rational number" or have any of

a variety of other meanings.)

As I was saying, in the simplest contexts

the dimension of a space is the number of

scalars it takes to specify a point in the

space. Often one simply identifies a point

in a space with a tuple of scalars; thus a

point in a plane is identified with a _pair_

of scalars, the ordinary x- and y-coordinates

that you were taught in high school; a point

in a three-dimensional space is idenitified

with a _triple_ of scalars, the x-, y-, and

z-coordinates that you may also have learned

in high school; a point in a four-dimensional

space is identified with a _quadruple_ of

scalars, which one could call w, x, y, z.

Et cetera..... If you like, you could say

the dimension of a space is the number of

quantities that can freely vary as you move

from one point to another within the space.

I gave the concrete example of longitude,

latitude, and altitude being three numbers

specifying a point in a three dimensional

space, used in case we're talking about the

vicinity of the surface of the earth.

Some complications: (1) The dimension of a

space does not depend on the particular system

of coordinates you're using, and yet I spoke

of the x-, y-, and z-coordinates above.

(2) This is just one of a number of related

senses that the word "dimension" has in

mathematics.

A simple example of the occurrence of a

five-dimensional space in the study of

ordinary old-fashioned concrete three-

dimensional Euclidean geometry: the space

of all shapes of tetrahedra is a five-

dimensional space.

An uncontroversial example from physics:

in statistical mechanics one works with

a space whose dimension is three times

the number of molecules of gas in a

chamber. Of course the number of

molecules is typically very large.

Some of the ordinary propositions of

Euclidean geometry, e.g. the Pythagorean

theorem, apply to this space with no

essential changes.

I also mentioned an example from statistics

above, although I didn't give any detail. 1174485622:::Mike Hardy:::hardy@math.umn.edu::::::I suppose I ought to answer Diana Hsieh, who

wrote:

: I have no interest in or even tolerance

: for his

[i.e., my]

: inquiries. Pat deserves better, particularly

: from someone without any first-hand knowledge

: of her work.

:

: Mike, haven't we already gone around on this

: issue on more than one occasion with respect

: to PARC?!? There's a principle involved here,

: a principle of justice.

What have I said about PARC? A summary:

(1) I read a section of about 50-or-60-or-so

pages of Ayn Rand's private journal with

occasional brief comments by James Valliant.

It confirmed the impression I got many years

earlier by reading Nathaniel Branden's book

_Judgment_Day_, which is that for Branden,

his NBI days were largely a power trip. The

"power trip" part seems to be Valliant's

bottom-line conclusion of that section.

(2) I also read less extensively from the

parts where Valliant indicts Barbara Branden's

book. I said here and elsewhere that the parts

I read looked as if they may identify some

errors in her book, but the ones I read about

didn't seem like the sort that can be explained

only by dishonesty.

(3) I also said I couldn't be sure even that he

had identified any errors because I didn't have

BB's book at hand. But now I do and I've also

bought a copy of Valliant's book, so I'll look

at the latter with the former at hand so I can

check.

(4) I've said some things that could probably be

summarized by saying I hope Valliant's wrong

about BB's book.

I never commented on parts of Valliant's book

that I haven't read.

So how is this unjust?

A few of the people who've posted here have

left me thinking that the part of this that

offends them is the hope that Valliant is

wrong about BB's book. That part blindsided

me; I didn't know that sentiment was out there.

Concerning Pat Corvini's work, I said I am

generally inclined to doubt anything will come

of claims to have solved problems in the

philosophical foundations of mathematics by

using the Objectivist epistemology, and this

is such a case, but nonetheless I hope Corvini's

got it right. I certainly never said she's

wrong. 1174486391:::Burgess Laughlin::::::http://www.aristotleadventure.com:::In Comment 39, Mike Hardy said: "I did not state a definition [...]"

We agree.

Mike: " [...] what I did state should be quite informative to anyone who didn't know that 'dimension' doesn't mean anything like the things proposed here."

Informative about what topic? I am having trouble following you. The two negatives ("... didn't ... doesn't ...") are too much for me to grasp with confidence. But I will set that aside. It isn't central to the problem -- which is identifying the *fundamental* nature of the referent units of the concept "dimension" as used in physics and mathematics or (possibly but doubtfully) in philosophy.

Mike: "Let's try a definition valid in at least some of the relevant contexts: [...]

Alright, I am now primed for a definition.

Mike: "[...] the dimension of a vector space is the maximum number of vectors in the space that can be linearly independent, and hence is the number of scalars needed to specify any one particular vector. [...]"

As far as I can tell, what you have offered is not a DEFINITION. You have instead offered an EXAMPLE application. If I were to ask -- What does "color" mean? -- and you were to say -- "The color of this apple is red" -- you would not have *defined* the concept "color." Instead you would have given an example. That is okay because identifying one unit subsumed by the concept is one of the many steps toward concept formation (and thus definition of the concept).

Mike: "More later...."

Good. I am looking forward to seeing a definition -- identifying the genus and differentia -- of the concept "dimension" as used in physics, mathematics or anywhere else, at this point.

P. S. -- For anyone interested in the meaning of "definition," within the context of Objectivism, you might study the entry for "Definition" in *The Ayn Rand Lexicon*, pp. 117-121. For somewhat more advanced study, see Ayn Rand's Ch. 5, "Definitions," *Introduction to Objectivist Epistemology* (about fifteen pages), plus the many entries under "Definition" in the index of *ITOE*. 1174491128:::Mike Hardy:::hardy@math.umn.edu::::::I wrote a definition:

:: [...] the dimension of a vector space

:: is the maximum number of vectors in the

:: space that can be linearly independent,

:: and hence is the number of scalars needed

:: to specify any one particular vector. [...]

Burgess Laughlin answered:

:: what you have offered is not a DEFINITION.

That is mistaken. What I wrote is indeed a

definition, and you'll find it in many standard

textbooks.

:: You have instead offered an EXAMPLE application.

Wrong.

:: If I were to ask -- What does "color" mean?

:: -- and you were to say -- "The color of this

:: apple is red" -- you would not have *defined*

:: the concept "color." Instead you would have

:: given an example.

Correct.

:: That is okay because identifying one unit

:: subsumed by the concept is one of the many

:: steps toward concept formation (and thus

:: definition of the concept).

I did not in fact give any example in the part

you quoted above. Ordinary Euclidean 3-space

with an origin chosen would be one of many

examples.

Really, I don't see how you're managing to

read what I wrote as an example.

And please not the later posting in which I

expanded on this further. 1174497304:::Fred Weiss:::fredweiss@papertig.com:::http://www.papertig.com:::Observing Mike Hardy foaming at the mouth and puzzled at the extent of his reaction, I finally realized that I may have partly contributed to the problem. That is, if he thinks I have a problem with the concept of "dimension", per se. But then all he would have had to do is ask me point blank, "What Weiss, you deny that physical objects exist in 3 dimensions?" But that's not what he did and instead we've gotten a lot of seemingly irrelevant ranting and overly technical expositions which seem more designed to display his wizardry rather than addressing the issue.

Frankly, at this point, I think it's clear that Mike is incapable of dealing with the question on the table.

For those of you who are interested in some of the issues surrounding string theory I refer you to the comments of Sheldon Glashow.

<http://www.pbs.org/wgbh/nova/elegant/view-glashow.html>

In contrast, there is Brian Greene, one of its leading proponents.

<http://www.pbs.org/wgbh/nova/elegant/greene.html>

Decide for yourself. 1174501491:::Burgess Laughlin::::::http://www.aristotleadventure.com:::Mike, I think we are making progress toward a resolution, one way or the other -- at least for my puzzle. Bear with me.

In Comment 43, you say: "I wrote a definition:

:: [...] the dimension of a vector space

:: is the maximum number of vectors in the

:: space that can be linearly independent,

:: and hence is the number of scalars needed

:: to specify any one particular vector. [...]

Burgess Laughlin answered:

:: what you have offered is not a DEFINITION.

That is mistaken. What I wrote is indeed a definition, and [...]"

Now we are at the crux of the problem. You can resolve it easily with simple answers to the following fill-in-the-blank questions:

1. In your "definition" above, the definiendum is what? ___________________.

2. What is the genus? __________________

3. What is the differentia? _________________

You continue: "I did not in fact give any example in the part you quoted above. Ordinary Euclidean 3-space with an origin chosen would be one of many examples.

Really, I don't see how you're managing to read what I wrote as an example."

Mike, I am bewildered here. Are you truly saying that the "dimension of a vector space" is not an example of a dimension?

I have repeatedly asked for a definition of the CONCEPT "dimension." You persist in avoiding the question and instead seem to want to describe instances of ideas (which in this case, by the way, are not CONCEPTS) such as "dimension of a vector space" or "vector space." Those ideas are compounds of concepts, but not themselves concepts, of course.

If I recall correctly, in an earlier post you said, in my words, that you doubt mathematics can be based on Objectivist epistemology. To make such a statement, of course, you must have thoroughly studied Objectivist epistemology. You must also, then, surely know the meaning of "concept." So I am at a loss in trying to explain why you *seem* to think that "dimension of a vector space" or "vector space" is a concept. I hope you will explain and clear up my confusion. 1174504776:::Mike Hardy:::hardy@math.umn.edu:::::::: Observing Mike Hardy foaming at the mouth and

:: puzzled at the extent of his reaction, I finally

:: realized that I may have partly contributed to

:: the problem.

No kidding. Sigh.

:: But then all he would have had to do is ask me

:: point blank, "What Weiss, you deny that physical

:: objects exist in 3 dimensions?" But that's not

:: what he did and instead we've gotten a lot of

:: seemingly irrelevant ranting and overly technical

:: expositions which seem more designed to display

:: his wizardry rather than addressing the issue.

What you call "irrelevant ranting and overly

technical expositions" to "display [my] wizardry"

were in fact directly addressing questions asked

by Burgess Laughlin.

They were responses to _Bugess_Laughlin_, not to _you_. 1174505369:::Mike Hardy:::hardy@math.umn.edu:::::::: 1. In your "definition" above, the definiendum

:: is what? ___________________.

::

:: 2. What is the genus? __________________

::

:: 3. What is the differentia? _________________

The definiendum is "dimension of a vector space".

That is indeed a concept.

It is NOT simply an example of a more general

concept of dimension. There are concepts such

as dimension of a manifold or Hausdorff dimension

of a subset of Euclidean space---there are a

fairly large number of such concepts. But they

are separate concepts. I won't deny the possibility

of some general concept subsuming all of them, that

would admit a definition of its own. And certainly

these are related concepts. And the dimension of

a _manifold_ is really what we'd need in order to

talk about the physics that started this whole

thread.

The genus could be taken to be integer-valued

functions assigning a value to each finite-dimentional

vector space.

And the differentia is something I was quite

explicit about earlier.

LEGAL NOTICE: This is a response to Burgess

Laughlin's question, not to anything that Fred

Weiss wrote. 1174508204:::Burgess Laughlin::::::http://www.aristotleadventure.com:::In Comment 47, Mike quotes me and responds:

":: 1. In your "definition" above, the definiendum

:: is what? ___________________.

::

:: 2. What is the genus? __________________

::

:: 3. What is the differentia? _________________

The definiendum is "dimension of a vector space".

That is indeed a concept."

Mike you may be pleased to note that this is probably my last post in this particular discussion. I apologize for having made an unwarranted assumption: That you know and agree with Objectivist epistemology, the core of which, of course, is Ayn Rand's theory of concepts.

Your claim that "dimension of a vector space" is a concept is false. "Dimension" is a concept (which I have repeatedly and fruitlessly asked you to define.) "Of" is a concept." "A" is a concept. "Vector" is a concept. "Space" is a concept. "Dimension of a vector space" is surely an idea of some sort, but it is not a concept.

Perhaps you simply do not understand Objectivist epistemology or perhaps you have understood it but don't agree with it. Either way, to continue the discussion would be pointless since we are operating in the context of different epistemologies. No wonder we had so much trouble communicating (or not communicating).

You are very intelligent and articulate. However, you need to either study closely Ayn Rand's theory of concepts or, if you have aleady done that and don't agree with it, then perhaps you might want to let people know that that is the situation so they won't assume you are an Objectivist just because you are posting in a generally Objectivist forum.

P. S. -- For anyone interested in knowing about what Ayn Rand means by "concept," I recommend a general, slow study of *Introduction to Objectivist Epistemology*. For a definition of "concept," see especially p. 10. For the use of a *single* word to symbolize a concept, see pp. 10, 40, and 174-177. 1174512251:::Fred Weiss:::fredweiss@papertig.com:::http://www.papertig.com:::Well, more basically, it's a fundamental principle of definitions that you don't use the term you are supposed to be defining in the definition.

If you are trying to define "dimension", you don't offer "dimension of a vector space" as the definition because obviously you still haven't answered the question "what is a dimension"?

But I completely agree with Burgess that for this and other reasons it is clear Mike is incapable of dealing with the issues on the table.

Beyond that I can't resist responding to this gem:

"What you call "irrelevant ranting and overly

technical expositions" to "display [my] wizardry"

were in fact directly addressing questions asked

by Burgess Laughlin."

I'm sure Burgess is thrilled to hear that your irrelevant ranting was directed at him. 1174555053:::Mike Hardy:::hardy@math.umn.edu::::::Fred Weiss, I am a professional and I am honest.

These are matters within my professional expertise

and unfamiliar to you.

I did not give a definition of "dimension", but

rather of "dimension of a vector space", because

I do not know a concept expressed only by the

word "dimension" except insofar as that is narrowed

down by context or by such a phrase as "of a vector

space". I answered Laughlin Burgess's questions,

thereby dealing with the issues he put on the table.

I did _not_ thereby deal with the issues you raise

and I did not intend to. And you are not Burgess

Laughlin's spokeperson.

You, Fred Weiss, are the beneficiary of my kindness

and are being dishonest.

Burgess Laughlin: You are wrong to think that only

things expressed by a single word can be concepts.

For example, awyers use the phrase "malice aforethought"

to denote, NOT a particular kind of "malice", but

a separate concept.

What in hell is it about some of the people posting

here that makes them think they can pontificate

like this about matters requiring study when they

haven't studied, and can so casually correct experts

who have? 1174563834:::Mike Hardy:::hardy@math.umn.edu::::::Burgess Laughlin wrote (almost...):

:: "Conceptual" is a concept.

:: "Common" is a concept.

:: "Denominator" is a concept.

:: "Conceptual common denominator" is

:: surely an idea of some kind, but it

:: is not a concept. For the use of

:: a *single* word to symbolize a

:: concept, see pp. 10, 40, and 174-177.

He also (almost) wrote:

:: "Sense" is a concept.

:: "Of" is a concept.

:: "Life" is a concept.

:: "Sense of life" is surely an idea

:: of some kind, but it is not a

:: concept.

OK, he didn't _quite_ write those things.

A couple of typos sneaked in somehow.

But you get the idea.

BTW, "vector space" is a concept that

cannot be understood simply by understanding

the words "vector" and "space", and indeed

"space" doesn't really denote any particular

concept when used in the phrase "vector space".

I _have_ seen it written as just one word:

"vectorspace". I haven't looked on those

cited pages of ITOE, but I would disbelieve

any assertion that Ayn Rand would have maintained

that "conceptual common denominator" is not

a concept she introduced in that book on the

grounds that it's expressed by more than one

word. 1174564886:::Mike Hardy:::hardy@math.umn.edu:::::::: I'm sure Burgess is thrilled to hear that

:: your irrelevant ranting was directed at him.

It was quite relevant to the questions he asked.

It was directed primarily _to_ him, and _to_

whoever else cares.

But look: you flunked math and you're

lying about that.

I will offer you my services as a tutor in

mathematics for $200 per hour if you promise

to study hard and you keep the promise. 1174565440:::Adrian Hester:::::::::Fred Weiss writes, "But I completely agree with Burgess that for this and other reasons it is clear Mike is incapable of dealing with the issues on the table." Unfortunately, neither is Fred Weiss; both sides are arguing about such different things that I don't see either side making itself understood to the other. First, a definition of "dimension" as used in physics and math was asked for, and one was provided. It was a specialized use of the more general term "dimension," true, but that doesn't change the fact that it defines "dimension" exactly in the context of vector spaces. You have to keep in mind that mathematics abstracts away from physical objects entirely to focus on the various *relationships* that can be found in the world. What Fred Weiss and Burgess Laughlin are asking for is essentially a definition of *spatial* dimensions, whereas Mike Hardy gave a definition of generalized dimensions suitable for abstract spaces. (The most general definition of "dimension" would abstract away from that context; I'll get to that a bit later.)

To understand the basis in reality for the abstract sense of dimension used in math and physics, start with the fact that there are three dimensions in space, length, breadth, and width; this is the sense of "dimension" that Fred Weiss and Burgess Laughlin are asking to be compared with the mathematical senses of the word. Now, to specify the position of a point, you need three numbers, the distance from a reference point in three distinct directions (usually as measured along three mutually perpendicular axes, though there are more general systems of axes you can use, such as the axes of a crystal whose axes aren't perpendicular, measurement along which is used to specify the position of a particular atom; this sort of space is of basic importance to crystallography). The basic property of these three numbers that is the basis for mathematical extensions of spatial dimensions is that the three numbers can in general vary independently of each other, and each set of numbers specifies a different point in space. This is the conceptual basis for graphs, such as a graph of your bank balance versus time. There are any number of possible values for your bank balance, from massive debt (negative values) to something on the order of the complete money supply of the United States, and the time varies throughout the period in which the bank account is open. Neither of these numbers is a measurement of a spatial dimension; instead, they are measurements represented by lengths or distances in physical space.

So what do these measurements and spatial measurements have in common? The fact that the numbers being graphed against each other can vary freely in such a way that different sets of these numbers specify different conditions. The complete set of all possible combinations of such numbers constitutes a conceptual or abstract space; each set of such numbers can be treated as a vector (an ordered set of numbers, each one a distinct measurement), in which case we're in the context of a vector space; and the number of such measurements is the dimension of the space. (More precisely, the smallest number of measurements uniquely defining a single point or situation. For example, you can specify a point on the surface of a sphere with its three coordinates in space, but these three numbers are not mutually independent; to specify the location of the point uniquely, you only need two numbers, longitude and latitude, say.) The most general definition of dimension in mathematics is the number of parameters (distinct values) necessary to uniquely specify all the possible situations (represented by points) in a given model, but not necessarily treating them as ordered sets of numbers. There's no limit on the number of dimensions in an abstract space, in this view, but these dimensions aren't spatial dimensions indicating measurement along a particular dimension either. This is where Fred Weiss's comment about string theory comes in: Arguably fundamental particles have to be specified in terms of 10 or 11 distinct measurements, but string theorists insist that these measurements are all spatial. 1174565612:::Adrian Hester:::::::::I wrote: "There's no limit on the number of dimensions in an abstract space, in this view, but these dimensions aren't spatial dimensions indicating measurement along a particular *dimension* either." That should read "measurement along [or in] a particular *direction*." 1174566147:::Mike Hardy:::hardy@math.umn.edu:::::::: The most general definition of dimension

:: in mathematics is the number of parameters

:: (distinct values) necessary to uniquely

:: specify all the possible situations

:: (represented by points) in a given model,

:: but not necessarily treating them as ordered

:: sets of numbers.

It seems rash to say that's the most general.

(I'm not prepared to offer an opposing candidate

though.) Everyone learns that one as an

undergraduate. Mathematics goes _way_ beyond

the undergraduate level. 1174566542:::Mike Hardy:::hardy@math.umn.edu::::::Geez... first time I've received off-list

fan mail as a result of posting here. 1174574457:::kishnevi:::jbennetsmith@hotmail.com::::::Re: Mr. Weiss in comment 37.

You are attributing a position to me that I'm not taking.

I'm beginning to feel that not only are studies in math needed by some people here,but also studies in basic reading comprehension.

You made a comment which effectively stated that "if it's not possible to perceive it, it does not exist".

I responded that the most you can legitimately claim is "if it's not possible to perceive it, it is not possible to talk about it." Assertions of existence and non existence of the unperceivable are both invalid. On this point, unless I need to take a course in reading comprehension myself, you seem to agree with me.

I did not demand that you prove the nonexistence of such things. I merely think your original statement was much too overbroad. 1174581858:::Fred Weiss:::fredweiss@papertig.com:::http://www.papertig.com:::Adrian, I greatly appreciated most of your comments, especially when you wrote, "There's no limit on the number of dimensions in an abstract space, in this view, but these dimensions aren't spatial dimensions indicating measurement along a particular dimension either. This is where Fred Weiss's comment about string theory comes in: Arguably fundamental particles have to be specified in terms of 10 or 11 distinct measurements, but string theorists insist that these measurements are all spatial."

I thought that was clear in my original comment about the "stolen concept" in string theory, as it was apparently to you - and to Burgess.

What I wrote was, "there cannot be another dimension...of reality. Is that other dimension real? Then it is part of reality, not "another dimension"."

Mike Hardy interpreted that to be a rejection of dimensionality, as such, including in mathematics and never once addressed my original point.

However my lack of knowledge of mathematics and physics - a lack which I openly acknowledged in previous comments leading up this - made it difficult for me to make the distinctions you did and which, I might add, you stated very clearly. Had I done so we might have avoided the rancorous exchange which ensued. 1174653439:::Mike Hardy:::hardy@math.umn.edu::::::Looking back over this page, I see that in

the early stages Fred Weiss made some quite

worthwhile comments. Then in post #30 he

became a complete crackpot, condemning without

bothering to understand. I then completely

forgot that he'd been reasonable earlier.

Next Burgess Laughlin tried to pin me down

on certain definitions, and complained that

they were NOT definitions, saying only after

going around and around that something named

by a phrase rather than a single word cannot

be a concept. "Conceptual common denominator"

therefore must not be a concept, but rather

some sort of idea to be understood only by

understanding three separate concepts, each

denoted by a word.

During that whole exchange, Fred Weiss said

repeatedly and correctly that I was neglecting

to answer him. When I said that was not

because of inability to answer him while

typing so many words, but rather because I

was answering Burgess Laughlin, he then said

I was directing my irrational "ranting" at

Burgess Laughlin. The fact that Burgess

Laughlin repeatedly requested more and more

of precisely that information from me

apparently doesn't exonerate me of directing

irrational ranting against him.

So it would seem that I neglected the fact

that Fred Weiss can make valid contributions.

All because his crackpot post #30 effectively

superseded that.

If anyone knows where to find Burgess Laughlin,

could you tell him I'm curious to know whether

my post #51 correctly represents his position. 1176188560:::Burgess Laughlin::::::http://www.aristotleadventure.com:::I last posted in this thread in comment 48, if I have counted correctly. I have not read any posts since then. Nor will I be returning. I have learned a lot from this experience, but I need to move on.

APOLOGY

I owe Mike Hardy and others an apology. I erred egregiously. I know that now, through the patient and clear explanations offered by mathematicians and a philosopher in another forum.

I am now convinced that "vector space," as used in the mathematical world, is *not* a qualified instance of a concept -- that is, "vector" does not qualify (delimit, subdivide) "space," and "space" does not qualify "vector." Rather, "space" appears to be context- or perspective-setting in some way.

I erred through ...

- Ignorance: (1) not *at all* understanding the idea of vector space, and (2) not *fully* understanding the idea of "qualified instance."

- Improper procedure: not first understanding how mathematicians are *in fact* using the term "vector space" to symbolize an idea.

As I understand it now, a true "qualified instance of a concept" (such as the QIC "dining table") would be a mental integration formed by subdivision from an earlier concept ("table") but similar enough to it in some type of measurement (height from the floor, for example) not to justify a single new word-symbol. "Desk," however, though a type (subdivision) of "table" in some respects, deserves a promotion to a new concept, symbolized by one word, in part because it requires a new, functionally significant line of measurement (storage capacity).

An apology should state the error, the proper course, and what one intends to do to correct or at least avoid repeating the impropriety. I have stated my errors and the proper answer, given my current and limited understanding. What I intend to do to avoid repeating my error is to be more cautious about reaching conclusions. This does not mean laymen shouldn't ask specialists questions, even persistently so, but it does mean that if the specialists don't immediately respond with clear and complete answers then that fact alone is not justification for jumping to conclusions about the invalidity of their assertions.

So, in other venues and on other topics, what I intend to do is proceed more slowly in such discussions and keep probing for objectivity.

I wish everyone the best,

Burgess Laughlin 1176392538:::Mike Hardy:::hardy@math.umn.edu::::::Thank you Mr. Laughlin---I appreciate your comments. 1177678936:::Jason Rheins:::::::::A plug for my course on Kant: Among other things, I will be discussing Kant's posit of a priori forms of cognition to explain the possibility of mathematics and mathematical physics. In this vein I will explain the traditional reasons for rejecting mathematical empiricism and discuss the challenges that a theory of empirical knowledge of mathematics must (and can) meet. Given the discussion above, that seems to be of much interest to people here. So, take the course.