May 20, 2002 11:16 AM Subscribe

Stephen Wolfram has finished his book, "A New Kind of Science," which purpotedly is being espoused as a paradigm shift in many fields. But, I'm starting to see a very reductionistic attitude in many of the main theorists of complextity theory and emergent phenomena. Is the idea that the Universe is in lines of code a phallus-extension/masculine overdriven idea? Isn't math a man made mapping and can the Universe be reduced to an equation by a man? Still this book is going to be groundbreaking. Read the following exceperpt from the wired.com article:
q: "I've got to ask you," I say. "How long do you envision this rule of the universe to be?"

w: "I'm guessing it's really very short."

q: "Like how long?"

w: "I don't know. In Mathematica, for example, perhaps three, four lines of code."

link via protofunk.org, old similar thread

posted by nakedjon (31 comments total)

w: "I'm guessing it's really very short."

q: "Like how long?"

w: "I don't know. In Mathematica, for example, perhaps three, four lines of code."

link via protofunk.org, old similar thread

posted by nakedjon (31 comments total)

The more I read about Wolfram (and it's interesting, I look forward to the book eagerly), the more I like the philosophies of Camus. Create your own meaning.

posted by atom128 at 12:15 PM on May 20, 2002

posted by atom128 at 12:15 PM on May 20, 2002

Yeah, and we'll probably have at least one more in a couple of months once somebody's managed to plough through it.

What if it is? That doesn't make it untrue and it doesn't make those lines of code less worth knowing.

posted by hob at 12:17 PM on May 20, 2002

Kurzweil rips him a new one.

Okay, it's not that bad, but he tries to point out some major holes in the idea.

posted by Su at 12:18 PM on May 20, 2002

Okay, it's not that bad, but he tries to point out some major holes in the idea.

posted by Su at 12:18 PM on May 20, 2002

Seriously now, not only has this been posted a zillion times, but I fail to understand what is either phallic or somehow 'male' about this idea. Superficially speaking, I would think it was the 1,100 page *book* that was 'phallic' or 'egocentric' or what have you -- not the compressed equation. And speaking more pointedly, isn't all of science, at least physics, reductionist in intent? And isn't that what Wolfram is trying to *undo*? As I understand it, his contention is that it is impossible to reduce certain complex behaviors into physical laws; he's saying that a 'few lines of code' generate 'complexity' within which behaviors occur unpredictably and spontaneously. He is generating *models*, not laws.

Is that reductionist (or even what his book is about)? I have no firm idea but it seems to me that if it is, it is different in character and intent from the 'reductionism' (or, as it were, 'science') practiced by others. Explain this phallo-centrism/reductionism/Wolfram link more. Has anybody read the book who can comment?

posted by josh at 12:18 PM on May 20, 2002

Is that reductionist (or even what his book is about)? I have no firm idea but it seems to me that if it is, it is different in character and intent from the 'reductionism' (or, as it were, 'science') practiced by others. Explain this phallo-centrism/reductionism/Wolfram link more. Has anybody read the book who can comment?

posted by josh at 12:18 PM on May 20, 2002

I bought this book yesterday (it was actually hard to find in NYC, finally spied it at St. Mark's Books) and read the first 45 pages.

The ideas are *quite* reductivist in a way very different than nakedjon is implying. Having read a whopping 3-4% of the book already I see that he is making the case for the Universe not being equations so much as algorithms. Very, very different things, these two. Algorithms are functional, procedural concepts and express something far different than an equation. You express an algorithm in some finite numbers of code-lines, but they have a life of their own, so to speak, and this book seeks to point out that it doesn't take a lot of complexity in the algorithm to generate a *lot* of complexity in the function/output of that algo. It's pretty cool stuff, despite Wolfram's universe-sized ego.

And, BTW:*"phallus-extension/masculine overdriven idea"*? I think that this is a pretty dumb way to dismiss an idea. Actually incredibly dumb.

And, Kurzweil is a complete nimrod. The man's books are jsut one big parade of fallacy... it makes my day complete that he is poking a hole in someone else's ideas. I've met him a couple of times and he jsut oozes self-absorbed bullshit to the point that I have to resist the temptation to make it my life's work to prove everything he says wrong. Not that Wolfram's going to turn out any different, mind you.

posted by n9 at 12:22 PM on May 20, 2002

The ideas are *quite* reductivist in a way very different than nakedjon is implying. Having read a whopping 3-4% of the book already I see that he is making the case for the Universe not being equations so much as algorithms. Very, very different things, these two. Algorithms are functional, procedural concepts and express something far different than an equation. You express an algorithm in some finite numbers of code-lines, but they have a life of their own, so to speak, and this book seeks to point out that it doesn't take a lot of complexity in the algorithm to generate a *lot* of complexity in the function/output of that algo. It's pretty cool stuff, despite Wolfram's universe-sized ego.

And, BTW:

And, Kurzweil is a complete nimrod. The man's books are jsut one big parade of fallacy... it makes my day complete that he is poking a hole in someone else's ideas. I've met him a couple of times and he jsut oozes self-absorbed bullshit to the point that I have to resist the temptation to make it my life's work to prove everything he says wrong. Not that Wolfram's going to turn out any different, mind you.

posted by n9 at 12:22 PM on May 20, 2002

Oh please.

This idea may or may not have merit; its merit will be determined by the same processes that determine the merit of any other scientific idea. The notion that the laws of physics are the

Unless there's a Wimmin's version of Newton's Law out there... F=2ma? I'd love to see the experiments...

posted by Hieronymous Coward at 12:22 PM on May 20, 2002

ok, I read the whole Kurzweil response. He doesn't rip anyone a new asshole. He just goes on and on about how Wolfram's book can't be all that important because it doesn't resemble Kurzweil's own thinking about strong AI, which has little to do with Wolfram's book. I guess Kurzweil thinks there is only enough room for one meglomanaical my-big-idea-will-change-everything kid on the block. Man, I feel all gross from reading that essay... Ray is just, man, such an intellectual *pig*.

his dismissals seem to follow along the track that Wolfram's 1-d and 2-d algos don't 'evolve' into AIs:

"...they are interesting (and intelligent) only to a degree. Moreover, they do not continue to evolve into anything more complex, nor do they develop new types of features. One could run these automata for trillions or even trillions of trillions of iterations, and the image would remain at the same limited level of complexity. They do not evolve into, say, insects, or humans, or Chopin preludes, or anything else that we might consider of a higher order of complexity than the streaks and intermingling triangles that we see in these images."

Ugh. Like he often does, he wholly missing the point. The book demonstrates the foundation of a different mode of thought. You don't dismiss Calculus because performing differentiation doesn't resemble driving a Porsche. Most of his responses seem shallow and full of context switching and fallacies. Wolfram might be arrogant and he may turn out to be wrong, but at least he's not as pompous a jerk as R.K.

posted by n9 at 12:46 PM on May 20, 2002

his dismissals seem to follow along the track that Wolfram's 1-d and 2-d algos don't 'evolve' into AIs:

"...they are interesting (and intelligent) only to a degree. Moreover, they do not continue to evolve into anything more complex, nor do they develop new types of features. One could run these automata for trillions or even trillions of trillions of iterations, and the image would remain at the same limited level of complexity. They do not evolve into, say, insects, or humans, or Chopin preludes, or anything else that we might consider of a higher order of complexity than the streaks and intermingling triangles that we see in these images."

Ugh. Like he often does, he wholly missing the point. The book demonstrates the foundation of a different mode of thought. You don't dismiss Calculus because performing differentiation doesn't resemble driving a Porsche. Most of his responses seem shallow and full of context switching and fallacies. Wolfram might be arrogant and he may turn out to be wrong, but at least he's not as pompous a jerk as R.K.

posted by n9 at 12:46 PM on May 20, 2002

Wolframfilter?

posted by davidgentle at 12:48 PM on May 20, 2002

posted by davidgentle at 12:48 PM on May 20, 2002

Is the idea that the Universe is in lines of code a phallus-extension/masculine overdriven idea?

Sure it is, in the same manner that anything that can be even remotely associated with any action that contains the most vague and/or obscure male attributes can be dismissed for that overdriven reason rather than being accepted or dismissed by it's own merits.

Quid pro quo.

posted by RevGreg at 1:14 PM on May 20, 2002

Sure it is, in the same manner that anything that can be even remotely associated with any action that contains the most vague and/or obscure male attributes can be dismissed for that overdriven reason rather than being accepted or dismissed by it's own merits.

Quid pro quo.

posted by RevGreg at 1:14 PM on May 20, 2002

What's the cultural, gender, and class makeup of mefi members anyway? Just kidding! The megalomaniac/egomaniac complex of many proprietors in different fields astounds me to the point of harkening warhol out of his grave. I'm more interested in how complexity theory is a description of a large network of intermingling parts and functions, yet it is one dude discussing the ideas and **specifically** addressing these ideas as his own. The wired article at least paints a picture of this guys taking the credit for all the people who are working on his book. I could see a book like this being written by a cast of characters like MeFi; an advanced network of social organization. But give me a break on this feudal lord commanding thunder. Its like hitler getting trippy on us again man.

**I am a big proponent of complexity theory and emergent phenomena though and of both Kurzweil and Wolfram's work and vision in the field.**

The interesting discussion about the masculine/feminine argument in relationship to science is that it is historically framed and continues in history to reduce the viewing range of the field. I mean come on, we map our entire concept of mathematics onto the world just like we [humans] map onto the Universe the concept of time and increments.

posted by nakedjon at 1:20 PM on May 20, 2002

The interesting discussion about the masculine/feminine argument in relationship to science is that it is historically framed and continues in history to reduce the viewing range of the field. I mean come on, we map our entire concept of mathematics onto the world just like we [humans] map onto the Universe the concept of time and increments.

posted by nakedjon at 1:20 PM on May 20, 2002

Those ideas aren't dismissed, but critiqued, discussed, and talked about. Just as Wolfram is probably trying to stir up a stagnant pond, why don't we jump in for a swim and mix it up some more [ie, phallus comment].

posted by nakedjon at 1:21 PM on May 20, 2002

posted by nakedjon at 1:21 PM on May 20, 2002

NakedJon, just so you know, mathematics exists independantly of us. We're not mapping math onto the universe, we're hoping to understand the universe through math. The math was there first. 2 is the only even prime whether or not there's anyone in the forest to observe it.

posted by bshort at 1:32 PM on May 20, 2002

posted by bshort at 1:32 PM on May 20, 2002

bshort:

Math exists in the heads of humans. Only the counting numbers can be said (arguably, imo) to be 'natural;' all other math constructs are abastractions and generalizations of them. ('1/2' only means anything if a 'whole' is defined, where in nature can you find an irrational? etc.)

*we're hoping to understand the universe through math.*

The formulas of physics are lossy compression of observable data. We are indeed mapping math onto the universe. There is not necessarily one special set of equations that can predict the entire universe; the universe may be incompressible (and if it is, it's certainly not compressible below it's kolmogorov complexity).

posted by sonofsamiam at 1:48 PM on May 20, 2002

Math exists in the heads of humans. Only the counting numbers can be said (arguably, imo) to be 'natural;' all other math constructs are abastractions and generalizations of them. ('1/2' only means anything if a 'whole' is defined, where in nature can you find an irrational? etc.)

The formulas of physics are lossy compression of observable data. We are indeed mapping math onto the universe. There is not necessarily one special set of equations that can predict the entire universe; the universe may be incompressible (and if it is, it's certainly not compressible below it's kolmogorov complexity).

posted by sonofsamiam at 1:48 PM on May 20, 2002

~laugh~

Tautologies R Us, eh? (That annoying sound you heard was an electric can opener liberating annelids.)

Sounds to me like you're still Waiting For Godel, and buddy, lemme give you a hint: he's already been here and gone.

posted by fold_and_mutilate at 1:56 PM on May 20, 2002

Einstein was known to say that he was skeptical if numeracy, that there was something 'off' about numbers being here first.

posted by n9 at 2:58 PM on May 20, 2002

posted by n9 at 2:58 PM on May 20, 2002

I am skeptical too. I am especially suspicous of how physicist often find that the mathematics for what they are trying to describe has already been invented. Riemann's geometry, matrix mechanics, Galois theory, Lie algebras were all mathematics invented as pure theory and later adopted as essential practical tools by physicists.

Its as if mathematics, as an extension of logic, tells us what sorts of things our mind can comprehend, lays out the basic templates. Later, in our encounters with reality, we search for and find or invent the particular "bucket" which can contain these new revelations.

posted by vacapinta at 3:24 PM on May 20, 2002

I made it 4 paragraphs into R.K.'s mindless essay. I gave up after I got tired of being talked down to. Aside from that, even with my sore lacking of deep scientific knowledge, I could see R.K. completely missing the point and just latching on to irrelevancies and addressing them -- and though he was getting to the 'heart' of the issue.

Clearly, Kurzweil likes to hear the sound of his fingers hitting the keys and loves to waste server-space with his tripe. We get the idea, you're a "smarty man". Now go away and leave discussion to those who address the relevant points and don't automatically assume we're right.

I can't stand people who take themselves too seriously.

posted by Dark Messiah at 3:37 PM on May 20, 2002

Clearly, Kurzweil likes to hear the sound of his fingers hitting the keys and loves to waste server-space with his tripe. We get the idea, you're a "smarty man". Now go away and leave discussion to those who address the relevant points and don't automatically assume we're right.

I can't stand people who take themselves too seriously.

posted by Dark Messiah at 3:37 PM on May 20, 2002

sonofsamiam: "The formulas of physics are lossy compression of observable data." Well, sure. And it turns out that’s exactly what Wolfram is saying. The smooth, continuous formulas that are so easy to predict don't necessarily crop up all that often in the real world. We work with them because they are really good shortcuts. By plugging in a few known variables we can see what values a smooth, continuous function has at any point in the given metric space. Wolfram is saying that some things can only be computed, not predicted. Oh, and, pi is pretty darn irrational.

Fold: "Sounds to me like you're still Waiting For Godel..." Godel's theorem puts restrictions on systems of logic and the resolvability of those systems into provable theorems. This turns out, for the most part, to be a mathematical curiosity, and doesn't hinder the existence of mathematical objects. For instance, the fact that Goldbach's conjecture is just a conjecture, and is not known to be provable does not hinder us from discovering other theorems and proving or disproving them.

posted by bshort at 4:41 PM on May 20, 2002

Fold: "Sounds to me like you're still Waiting For Godel..." Godel's theorem puts restrictions on systems of logic and the resolvability of those systems into provable theorems. This turns out, for the most part, to be a mathematical curiosity, and doesn't hinder the existence of mathematical objects. For instance, the fact that Goldbach's conjecture is just a conjecture, and is not known to be provable does not hinder us from discovering other theorems and proving or disproving them.

posted by bshort at 4:41 PM on May 20, 2002

I could see R.K. completely missing the point and just latching on to irrelevancies and addressing them

Kurzweil seems overly excited by the concept that complex things are made up of lots of simple things - and that there is an underlying logic to the behaviors of these simple things that is correspondingly simple, yet these simple behaviors when combined produce more complex behaviors.

Wow. What a concept.

posted by RevGreg at 4:51 PM on May 20, 2002

Kurzweil seems overly excited by the concept that complex things are made up of lots of simple things - and that there is an underlying logic to the behaviors of these simple things that is correspondingly simple, yet these simple behaviors when combined produce more complex behaviors.

Wow. What a concept.

posted by RevGreg at 4:51 PM on May 20, 2002

Rev: are you agreeing with me, or not? I honestly can't tell and, as such, am unsure how to proceede. Help me, my feeble brain is failing me!!

posted by Dark Messiah at 6:00 PM on May 20, 2002

posted by Dark Messiah at 6:00 PM on May 20, 2002

E = mc^{2}

That's pretty simple, and it has a lot of implications. I can believe four lines.

posted by mstillwell at 6:30 PM on May 20, 2002

That's pretty simple, and it has a lot of implications. I can believe four lines.

posted by mstillwell at 6:30 PM on May 20, 2002

Believe me, there is *nothing* more tiresome than a discussion of science/objectivity vs. social theory/subjectivity. *Nothing*. I come to MeFi to get away from my college courses, not to relive their worst, most nightmarish components. Ugh.

The fact is that, as one would expect, the physical sciences and social theory*do not* collide except when they are misinterpreted or generalized outside of their proper spheres. No one makes this clearer than Alan Sokal did with his hoax article in *Social Text*. It might be reasonable enough to talk about how psychology, say, has been influenced by social facts. But it is not sensible to talk about how physics has been influenced (and I think you would be hard-pressed to find an actual example of a development in physics that has had to be changed because of changing social ideas -- the heliocentric theory of the Universe, for instance, might have been inspired by the Bible, but was tossed because of observable data, not social change).

Nakedjon, if you think**this** particular set of theories is somehow 'phallocentric' I would be curious to hear why, but if there's no reason other than "science is masculine" then I don't think there is a discussion to have. Read Sokal's hoax article and tell me it doesn't sound pretty close to the kind of unusable position you're trying to take. (I'm a humanities kid too -- I've been known to *read* Social Text *on my own time!*)

posted by josh at 7:47 PM on May 20, 2002

The fact is that, as one would expect, the physical sciences and social theory

Nakedjon, if you think

posted by josh at 7:47 PM on May 20, 2002

(putting on my egotistical Wolfram hat)

What I haven't heard anyone say is that it's not so reductionist to say that the universe can be summed up in 'three or four lines of Mathematica code.'

Mathematica, in its entirety, is intended to be an algorithm-cruncher, and it's hardly a small program. To say that the universe can be represented in a small amount of code interpreted by that program is to say that the program is already inheirently similar to the universe, and three or four lines of tweaking or 'direction' if you will, is all that is needed to bring the hundreds of thousands of lines of code already written into alignment with our particular universe.

If Wolfram so desired, he could factor those in to the program itself until it only took a few characters to set the right algorithm in motion.

What I (and I believe Wolfram) am(is) saying is not that the universe is as simple as 3 or 4 lines, but that Mathematica already operates on rules and principles very close to the universe.

That said, my book's arriving tomorrow. The first printing run was only 50,000 copies, costing Wolfram $12 apiece because he was so picky about production methods. This morning the book was Amazon's #1 bestseller. This afternoon it was already dropping because estimated delivery times changed from 24 hours to two weeks (and likely far more, waiting for the second printing).

I love mathematica, and I like books that make me think. Whether this is the next revolution in science or just another Pons and Fleishman, I'm sure it'll make me think, and possibly give me a few insights into the power of Mathematica. $45 well spent.

posted by kfury at 8:47 PM on May 20, 2002

What I haven't heard anyone say is that it's not so reductionist to say that the universe can be summed up in 'three or four lines of Mathematica code.'

Mathematica, in its entirety, is intended to be an algorithm-cruncher, and it's hardly a small program. To say that the universe can be represented in a small amount of code interpreted by that program is to say that the program is already inheirently similar to the universe, and three or four lines of tweaking or 'direction' if you will, is all that is needed to bring the hundreds of thousands of lines of code already written into alignment with our particular universe.

If Wolfram so desired, he could factor those in to the program itself until it only took a few characters to set the right algorithm in motion.

What I (and I believe Wolfram) am(is) saying is not that the universe is as simple as 3 or 4 lines, but that Mathematica already operates on rules and principles very close to the universe.

That said, my book's arriving tomorrow. The first printing run was only 50,000 copies, costing Wolfram $12 apiece because he was so picky about production methods. This morning the book was Amazon's #1 bestseller. This afternoon it was already dropping because estimated delivery times changed from 24 hours to two weeks (and likely far more, waiting for the second printing).

I love mathematica, and I like books that make me think. Whether this is the next revolution in science or just another Pons and Fleishman, I'm sure it'll make me think, and possibly give me a few insights into the power of Mathematica. $45 well spent.

posted by kfury at 8:47 PM on May 20, 2002

Totally kfury...that was the reason I brought up this topic. I admire Wolfram's embracing of conflict to tweak some noggins and get people to take self-organize into a new structure to support a new science.

Thanks for the article josh : ) . Um, the argument of gender typing of roles was not meant to be a "one liner," or reduction of the man wolfram (just as i said this idea seemed). The phallic comment had more to do with the way Kurzweil and Wolfram talk about themselves as if it is the truth (implying, so don't question it).

posted by nakedjon at 10:50 PM on May 20, 2002

Thanks for the article josh : ) . Um, the argument of gender typing of roles was not meant to be a "one liner," or reduction of the man wolfram (just as i said this idea seemed). The phallic comment had more to do with the way Kurzweil and Wolfram talk about themselves as if it is the truth (implying, so don't question it).

posted by nakedjon at 10:50 PM on May 20, 2002

mathematical objects. For instance, the fact that Goldbach's conjecture is just a conjecture, and is not known to be provable does not hinder us from discovering other theorems and proving or disproving them.

Prove them? I'm with ya, friend. But do tell within exactly

If my hero Paul Erdos'

Behold living things, matter that contemplates itself -- recursive little bastards, all. That contemplation includes mathematics.

Well, I'm a big fan of both Mathematica (despite hating the licensing restrictions...consider Reduce or even MacSyma if you're looking for tools) and emergent computation/physics/cognition too, but having played with the sample programs from the book, I think you'll find they don't actually express much of what Mathematica is really designed to do, any more than an implementation of CAs in C++ will really turn you on to that language. The programs are fascinating stuff in their own right, however. I'm enjoying the book...shades of Fredkin and Feynman and John Walker Wheeler at their most speculative.

posted by fold_and_mutilate at 11:24 PM on May 20, 2002

Oh geez.... He never says that the Universe is in lines of code, just that there are simple rules that drive the complexity of the universe, and makes the comparison to Mathematica to indicate relative size of a ruleset in a complex system... Now how that has anything to do with phalluses is beyond me...

Additionally, I wonder why you complain about penis-something, then in the next breath, give man (not woman) all the credit for mapping math (though to what isn't clear). The universe itself

Just cause we know the rules to the game of Life (conway), doesn't mean we can predict what the 100th generation will like like without (a) having observed the same initial pattern in the past and watched it for 100 generations, or (b) completely and correctly modeling the Life universe and simulating the generations with the

posted by hurkle at 8:52 AM on May 21, 2002

bshort: *Well, sure. And it turns out that’s exactly what Wolfram is saying. The smooth, continuous formulas that are so easy to predict don't necessarily crop up all that often in the real world. We work with them because they are really good shortcuts. By plugging in a few known variables we can see what values a smooth, continuous function has at any point in the given metric space.*

That's all it is.*A really good shortcut.*

Just because you have a formula that describes a physical occurance as accurately as your instruments can measure does*not* mean that this formula is equivalent to whatever process actually governs the occurance. (and re: continuity, it's not clear that the universe is continuous.) It may be that the formulas that describe the universe's behavior precisely take up an entire universe.

Re: pi, where can you show me a Euclidean circle? At some resolution (atoms and below), it will break down ;)

But foldy has put it better than me.

posted by sonofsamiam at 9:56 AM on May 21, 2002

That's all it is.

Just because you have a formula that describes a physical occurance as accurately as your instruments can measure does

Re: pi, where can you show me a Euclidean circle? At some resolution (atoms and below), it will break down ;)

But foldy has put it better than me.

posted by sonofsamiam at 9:56 AM on May 21, 2002

I find it quite amusing that this post starts out with what can only be seen as a political attack on material which the poster could hardly have yet digested... this 1200 page brick has been out for all of a week.

Get this nugget: complexity isn't nearly as complex as everyone so far as made it to be. That's the starting point of the book. You don't need to a turing machine with 200 different 'states' in order to generate a complex system from a basic set of rules... just 5. Wolfram himself is man enough to admit that he once missed that particular point in his early analysis of Rule 30.

What politics could possibly have to add to this debate at this point I have yet to fathom, but then again I left college after a BA because I saw the field I was involved it devolve into ritualistic navel-gazing at the graduate level.

posted by clevershark at 10:45 AM on May 21, 2002

Get this nugget: complexity isn't nearly as complex as everyone so far as made it to be. That's the starting point of the book. You don't need to a turing machine with 200 different 'states' in order to generate a complex system from a basic set of rules... just 5. Wolfram himself is man enough to admit that he once missed that particular point in his early analysis of Rule 30.

What politics could possibly have to add to this debate at this point I have yet to fathom, but then again I left college after a BA because I saw the field I was involved it devolve into ritualistic navel-gazing at the graduate level.

posted by clevershark at 10:45 AM on May 21, 2002

Fold: "Prove them? I'm with ya, friend. But do tell within exactly which system you, a human, have chosen to prove them, or even use them."

See, I think you're looking for some sort of external confirmation of proofs. Like looking for spirals in sunflowers or chaotic behaviour in ant trails. I'm not saying that all mathematical objects exist in the real world. On the contrary, I think that mathematical objects (and grammatical object) could better termed to be abstract; i.e. non-spatial and non-temporal. Your insistence on the independant existance of math for it to be valid or universal doesn't really work. Because mathematical (and, indeed, formal) truths are forms of*a priori* knowledge, they are necessarily objective. If I say 1 = 1 or 2 = 1 + 1, those are true regardless of what you choose to believe. You could insist that mathematics is simply the formal manipulation of signs, but that would imply that different forms of notation would somehow have different truths.

sonofsamiam: "Just because you have a formula that describes a physical occurance as accurately as your instruments can measure does not mean that this formula is equivalent to whatever process actually governs the occurance."

Well, true. The formula (or the automaton) is not the same as the physical reality of processes, but you could perhaps argue that it is computationally equivalent, and therefore equivalent in the abstract.

"Re: pi, where can you show me a Euclidean circle? At some resolution (atoms and below), it will break down ;)"

As I was saying to fold, mathematical objects are abstract. Pi exists independantly of its approximate expression in nature. After all, if pi was exactly equal to the resolution of our measurement not only would it vary depending on the local curvature of space, but it would vary depending on the scale (an aside: to use it in formulae we would have to specify pi as a circle inscribed on such and such material that is such and such size measured to such and such resolution with a specific measuring device) that we're talking about. Additionally, pi would necessarily be limited by the Planck length, it would vary at a sub atomic level, etc.

posted by bshort at 1:11 PM on May 23, 2002

See, I think you're looking for some sort of external confirmation of proofs. Like looking for spirals in sunflowers or chaotic behaviour in ant trails. I'm not saying that all mathematical objects exist in the real world. On the contrary, I think that mathematical objects (and grammatical object) could better termed to be abstract; i.e. non-spatial and non-temporal. Your insistence on the independant existance of math for it to be valid or universal doesn't really work. Because mathematical (and, indeed, formal) truths are forms of

sonofsamiam: "Just because you have a formula that describes a physical occurance as accurately as your instruments can measure does not mean that this formula is equivalent to whatever process actually governs the occurance."

Well, true. The formula (or the automaton) is not the same as the physical reality of processes, but you could perhaps argue that it is computationally equivalent, and therefore equivalent in the abstract.

"Re: pi, where can you show me a Euclidean circle? At some resolution (atoms and below), it will break down ;)"

As I was saying to fold, mathematical objects are abstract. Pi exists independantly of its approximate expression in nature. After all, if pi was exactly equal to the resolution of our measurement not only would it vary depending on the local curvature of space, but it would vary depending on the scale (an aside: to use it in formulae we would have to specify pi as a circle inscribed on such and such material that is such and such size measured to such and such resolution with a specific measuring device) that we're talking about. Additionally, pi would necessarily be limited by the Planck length, it would vary at a sub atomic level, etc.

posted by bshort at 1:11 PM on May 23, 2002

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posted by gordian knot at 11:32 AM on May 20, 2002