Philosophy Reading Group / Kant & Math

[kote]

In order to promote the informed discussion of philosophy, I thought it might help if we could all be on the same page on some of the issues, literally. I would like to invite suggestions on relatively accessible philosophy texts that we could read and discuss. What issues have been on your mind? What can I read to get up to speed so we can have an informed discussion on the topic? If you are unsure of what text is out there, feel free to just ask about an issue and I can probably suggest or find a relevant article.

Hopefully a few people at least will get involved and we can share our thoughts and generate public discussion on the board. Obviously this would just be an unofficial and informal group.

Although I would really rather respond to inquiries or suggestions, let me suggest a text and initial question to get the ball rolling. It can't hurt to start with a classic, so how about Kant's "The Critique of Pure Reason?" I don't want to scare anyone off, so let's start with a single page in the text - Introduction Part V.1. It's page 19-20 here: The Critique of Pure Reason (http://www.e-text.org/text/Kant%20Immanuel%20-%20The%20Critique%20of%20Pure%20Reason.pdf). Specifically, I'll start with a question on the quoted section below (but please read the rest of the section at least).
We might, indeed at first suppose that the proposition 7 + 5 = 12 is a merely analytical proposition, following (according to the principle of contradiction) from the conception of a sum of seven and five. But if we regard it more narrowly, we find that our conception of the sum of seven and five contains nothing more than the uniting of both sums into one, whereby it cannot at all be cogitated what this single number is which embraces both. The conception of twelve is by no means obtained by merely cogitating the union of seven and five; and we may analyse our conception of such a possible sum as long as we will, still we shall never discover in it the notion of twelve. We must go beyond these conceptions, and have recourse to an intuition which corresponds to one of the two− our five fingers, for example...Do you agree with Kant here? Do you agree that our conception of the number 12 depends to some degree on experience and isn't something we can derive rationally?

Please feel free to jump in. No other background is necessary, although the first parts of the text could be helpful to read. Play along on this one and then suggest your own. It could be a book or a recent article or paper - anything. Let's just try to concentrate on one at a time to focus discussion and allow time for reading.

[kote]

And to start it off... I personally don't buy it. I'm of the belief that we don't need to learn anything from our environment to know that 12=7+5. We can figure this out completely from the definitions of the words without visualizing anything.

Can this be proven though? The criteria he uses make it tricky. It almost turns into a problem of psychology. Are we capable of counting before we see or feel anything? I think this is the wrong direction to go when we are discussing necessary vs contingent truth. Psychology can only explain the strength and source of our belief, not its truth. Or is psychology the only resource we can look to here?

[JoeDawg]

I think math is a way of generalizing experience. Before you can get to 1+1=2, you need to understand numbers are abstractions, a second order abstraction, that is different from a physical object (first order abstraction). You can apply '1' to any object, and group objects according to categories. However, once you have a conceptual framework built up that corresponds to the world of objects, you can use logic to gain new knowledge, but only based on how accurate your framework is, at representing those real world objects.

Even real world objects are abstractions, mainly based on their utility or lack thereof. (This is not an ontological statement about their physical nature, but rather how we relate to what is out there)
A rock is something we can pick up, a bunch of rocks we can count, lots of rocks, which we can't be bothered to count, is 1 pile, a lot more is 1 hill.... or 1 mountain...etc..

However, abstractions, first or second order, don't always correspond to the external world. I don't think you can get to 1+1=2, in any useful way, without real world experience for correspondence or validation. Your system could include it, but it could also include 1+1=3. Experience gives abstraction truth value.

Fortunately for mathematicians, our current conceptual framework has been built over centuries, designed and refined to correspond to observable phenomena.

Unfortunately, our conceptualization of real world objects is based on a limited evolved perspective. Which is why quantum and cosmic logic tend to boggle our brains. Even on the everyday level, our conceptual framework of objects can fail us, fail to correspond, to meaningfuly describe the world. And to a certain degree, since math is more abstract, but also more rigourous, math leads to fewer errors than our more mundane object framework.

Sorry, thats not well organized. But I'm trying to keep it as simple as possible.

[TheStatutoryApe]

I fully agree. We need to learn the language before we can 'speak' and understand it. With out knowing '1+1=2', '1+2=3', ect there is no means of deducing the answer. Even the process needs to be experienced before it can be understood.

[0xDEADBEEF]

...
Although I would really rather respond to inquiries or suggestions, let me suggest a text and initial question to get the ball rolling. It can't hurt to start with a classic, so how about Kant's "The Critique of Pure Reason?" I don't want to scare anyone off, so let's start with a single page in the text - Introduction Part V.1. It's page 19-20 here: The Critique of Pure Reason (http://www.e-text.org/text/Kant%20Immanuel%20-%20The%20Critique%20of%20Pure%20Reason.pdf). Specifically, I'll start with a question on the quoted section below (but please read the rest of the section at least).
We might, indeed at first suppose that the proposition 7 + 5 = 12 is a merely analytical proposition, following (according to the principle of contradiction) from the conception of a sum of seven and five. But if we regard it more narrowly, we find that our conception of the sum of seven and five contains nothing more than the uniting of both sums into one, whereby it cannot at all be cogitated what this single number is which embraces both. The conception of twelve is by no means obtained by merely cogitating the union of seven and five; and we may analyse our conception of such a possible sum as long as we will, still we shall never discover in it the notion of twelve. We must go beyond these conceptions, and have recourse to an intuition which corresponds to one of the two− our five fingers, for example...Do you agree with Kant here? Do you agree that our conception of the number 12 depends to some degree on experience and isn't something we can derive rationally?...

No this is one of the places where Kant is obviously false. He splits the world into the synthetic and analytic statements, but let me answer the math part first. Kant argues that the concept of 12 is synthetic. By now we have shown that we can derive most of mathematics from a small set of axioms. The english word "implies" shows nicely that, whatever we derive as a result from the axioms was already present. So mathematics is analytical, and every proof is a consequence derived from the axioms, we don't create new mathematics, but we discover it. We have to excuse Kant here, because he came long before Peano, and didn't really grasp the nature of numbers like we do now.
Apart from bragging about all the things he knows about philosophy, I think there was another point to the chapter that you are reading, but I find it hard to put into words. If you take some simple pseudophilosophical statement like "Everything is love." Then Kant is annoyed by arguments that take something like this and answer unrelated questions with it. So say "Is capital punishment acceptable?" and a person says "In our great love for society it is our duty to protect it by killing those who do harm to it." What this person is doing is taking his concept of love and elaborating on what he understands love to be. But Kant says there is no new information in this. The person is just doing circles around his concept of love. Where he should really produce new knowledge with synthetic statements by connecting new things to the concept of love, not by searching for thing already contained in that concept.

Well at least that's how I read Kant...

[JoeDawg]

By now we have shown that we can derive most of mathematics from a small set of axioms.

Most?

[Sorry!]

Hey first kudos to you for starting this. I think it's a great idea and I hope that it is successful.

I find this view of mathematics pretty interesting. I assume with it math is no longer transcendental... or at least we can not say that the math we use is the same as the math an alien species would use.

I think that I believe sort of a middle-road. I think mathematics isn't a human cognitive phenomena, that it exist seperate of us but we can only understand it through experiences. There's no doubt that 1+7=8 is a purely human statement however the concept is universal. That is to say that the numbers the functions etc are given names and definitions by humans but without these names and without even humans minds if one rock joins with another rock there are more than originally, this will never change.

[0xDEADBEEF]

Most?

The problem is really Gödel. First order logic explains a lot, but once we use recursion we are troubled. Many types of strange loops will lead to valid math like the group A of all groups containing the group A, whereas the group B containing all the groups not containing B will not. Mathematics still doesn't handle this elegantly (yet) and there are still a few things out of reach but f.a.p.p. mathematics derives from our basic axioms.

[JoeDawg]

The problem is really Gödel.

A pretty big problem from what I have read. I think you're idealizing mathematics, like the ancient greeks did with geometry. It all happens in the human brain. And different cultures have developed different mathematical systems. Ours has simply absorbed all the aspects we find most useful.

Math is in large part analytic, but its axioms and basic logic are derived from experience. All logic comes from how we see the world working. Math is just a way to represent and predict experience using highly abstract language.

Axioms are little more than assumptions, or constraints. And those constraints are based on our experiences in the world.

[JoeDawg]

if one rock joins with another rock there are more than originally, this will never change.
And if you smash em together really hard you get lots of rocks.

[Sorry!]

And if you smash em together really hard you get lots of rocks.

You my friend, have just discovered division.

[JoeDawg]

You my friend, have just discovered division.
Only if one rock shatters.

If both shatter, you have addition and division.

And if you heat those pieces enough, 50 + 50 = 1

:)

[Sorry!]

Only if one rock shatters.

If both shatter, you have addition and division.

And if you heat those pieces enough, 50 + 50 = 1

:)

I'm taking notes no worries

[JoeDawg]

I'm taking notes no worries
Uhm... Ok.

[0xDEADBEEF]

A pretty big problem from what I have read. I think you're idealizing mathematics, like the ancient greeks did with geometry. It all happens in the human brain. And different cultures have developed different mathematical systems. Ours has simply absorbed all the aspects we find most useful.

Math is in large part analytic, but its axioms and basic logic are derived from experience. All logic comes from how we see the world working. Math is just a way to represent and predict experience using highly abstract language.

Axioms are little more than assumptions, or constraints. And those constraints are based on our experiences in the world.

Omnium rerum homo mensura est.
Well I beg to differ from you, Protagoras and all the evil Sophists. It is true, that we do not do math at random. The definition of a continuous function is motivated by our experience in the world. Nonetheless everything that was proved about it, has always been true, had we invented the concept or not. Maybe I'll borrow from Kant and call logic an a priory knowledge or go more towards Platonian ideas and say that mathematics exists irrespective of formulation. But the idea that mathematics is invented rather than discovered, is dangerous. Same goes for science as this snippet from "Surely you are joking Mr. Feynman" should illustrate:

One guy in a uniform came to me and told me that the army was glad that
physicists were advising the military because it had a lot of problems. One
of the problems was that tanks use up their fuel very quickly and thus can't
go very far. So the question was how to refuel them as they're going along.
Now this guy had the idea that, since the physicists can get energy out of
uranium, could I work out a way in which we could use silicon dioxide --
sand, dirt -- as a fuel? If that were possible, then all this tank would
have to do would be to have a little scoop underneath, and as it goes along,
it would pick up the dirt and use it for fuel! He thought that was a great
idea, and that all I had to do was to work out the details.

[JoeDawg]

Well I beg to differ from you, Protagoras and all the evil Sophists.

LOL

Fortunately I am in a time and place where I can't be burned at the stake for having evil ideas that go against doctrine. Funny thing about those sophists, one of the reasons Socrates said they were such bad people was they charged for their services, much like academics do today. Socrates would not be a fan of copyright. But some say the great teacher of Plato was really a sophist at heart, he was just fed by ego. Wisdom is knowing you know nothing, after all.

If you can show how math exists, the stuff that it is made up of, that is separate from the human mind, then you might be different from Pato, who believed in a fantastical higher realm of reality where his forms existed. Otherwise, defining math as discovered is like defining ice cream as such; it was always there, someone just had to find it.

And Feynman is talking about physics (science), which deals directly with descriptive empirical facts, not just with math, which is an abstraction from empirical fact. I did try and address the importance of this earlier. There is quite a history of trying to reconcile empiricism and rationalism, something that most recently the Logical Positivists tried and failed to do.

Even so, science advances when people have crazy ideas and then make them work. Math works whether it describes reality or not, its simply more useful when it does, and its the useful math that people get charged to learn.

But I like to live on the edge, and its been so long since I've been slandered with the title sophist, I'm quite amused, so you are in no danger from me. For the record, I don't charge, and I'm more into learning than teaching.

[apeiron]

Maybe I'll borrow from Kant and call logic an a priory knowledge or go more towards Platonian ideas and say that mathematics exists irrespective of formulation.

Unless you just flatly reject the idea that all knowledge is modelling, then we should be careful of applying the same words to describe what stands on either side of the epistemic gap.

So it is dangerous to say I have mathematical ideas and reality is also mathematical. By this collapse of jargon, you are eliminating the last essential vestige of doubt that keeps us epistemologically honest.

So why not say I have mathematical ideas and reality seems to have deep patterns? And so I can hope that, with care and testing, my mathematical ideas will come asymptotically close to capturing the "truth" of those deep patterns?

Those patterns would of course exist irrespective of whether my formulations are very good or perhaps quite partial or faulty.

So don't say A = A, but rather A = A'.

One claims an identity, which leads to endless arguments about: well how do you actually know? The other claims only a mapping. And the proof then is in the pudding.

[0xDEADBEEF]

I prepared a long answer, but my browser ate it...
Here the short form:
There is a problem with existence, which I tried to attack earlier by introducing a new word for the patterns you are mentioning, because I wouldn't say that they exist the same way as chairs do. I said they "besist", but it didn't find support.

The main conflict which I deem to be irresolvable is this:
You can argue that logic works the way it does, because it is how our brains work. So logic depends on the mathematical patterns of physics that the universe displays. But on the other hand we are minds, our discussion is based on logic (mostly :wink:), so we can argue that logic needs to be established before we analyze nature, because our minds run on it.

Well let me correct this, it's what my mind runs on, whereas you are just a peculiar physical phenomenon, to talk to on a web forum :tongue2:

[Sorry!]

I prepared a long answer, but my browser ate it...
Here the short form:
There is a problem with existence, which I tried to attack earlier by introducing a new word for the patterns you are mentioning, because I wouldn't say that they exist the same way as chairs do. I said they "besist", but it didn't find support.

The main conflict which I deem to be irresolvable is this:
You can argue that logic works the way it does, because it is how our brains work. So logic depends on the mathematical patterns of physics that the universe displays. But on the other hand we are minds, our discussion is based on logic (mostly :wink:), so we can argue that logic needs to be established before we analyze nature, because our minds run on it.

Well let me correct this, it's what my mind runs on, whereas you are just a peculiar physical phenomenon, to talk to on a web forum :tongue2:

If i recall the reason you didn't find support is because you want to invent a word to describe something we already have words for... whats the use?

As well what do you mean by this whole 'we are minds' stuff... I don't follow.

[apeiron]

The main conflict which I deem to be irresolvable is this:
You can argue that logic works the way it does, because it is how our brains work. So logic depends on the mathematical patterns of physics that the universe displays. But on the other hand we are minds, our discussion is based on logic (mostly :wink:), so we can argue that logic needs to be established before we analyze nature, because our minds run on it.

:

If two things are inextricably linked - like models and worlds - then you can't get one right ahead of the other. Instead it is all about the working relationships that develop.

Brains and realities have developed a relationship spanning circa 600 million years. So they have a wired in logic - which is dichotomistic. Figure-ground, attention-habit, etc.

Humans with language invented a more partial (if locally penetrating) logic in syllogistic reasoning and other analytic methods. This is not wired into the brain but a socially constructed habit. It is still "logic", but a subset of what we actually use - and what the universe also "uses".

[0xDEADBEEF]

If i recall the reason you didn't find support is because you want to invent a word to describe something we already have words for... whats the use?
To tell people not to say that prime numbers exist. Because they don't exist like a piece of bread.

As well what do you mean by this whole 'we are minds' stuff... I don't follow.

So who am I communicating with, if it is not you, or more exactly your mind. If we follow standard philosophy, we cannot say much about the world, except that it appears to us in a certain way. Nothing is sure except your perception of yourself. If we use this as a starting point then logic is already present. It is not (by necessity) a product of how the things which you perceive behave.

[apeiron]

Nothing is sure except your perception of yourself. If we use this as a starting point then logic is already present. It is not (by necessity) a product of how the things which you perceive behave.

While its true that it all boils back to cartesean doubt and the thought that it is only our own thinking that we can be sure of, there are still important alternatives here.

You are thinking that the thinking has ontic "being". That is exists. Or maybe besists! But I may feel that my thinking only persists. That it is a process. And so open to development.

On cartesean grounds, how can we decide who is right? Whose idea of logic correctly applies?

Well hang around a while and a "being" that is truly thinking will tend to notice development actually happening in their thought processes. The modelling relationship will emerge into view. Whereas the alternative, the thinking being - the soul-like essence with platonic-like rationality - never does. Even though a lot of brains have been diced, probed and scanned in the search.

So yes. Cartesean doubt (or Humean correlation) is the correct place to anchor scholarship. It is the acid test of epistemology. But then you get on with things and do the work that allows you to leave the doubt well behind.

You doubt everything to find the direction that then can systematically reduce all doubts to their practical limit.

[JoeDawg]

As well what do you mean by this whole 'we are minds' stuff... I don't follow.
Standard confusion about the difference between epistemology and ontology, I'm guessing.

[kote]

There is a problem with existence, which I tried to attack earlier by introducing a new word for the patterns you are mentioning, because I wouldn't say that they exist the same way as chairs do. I said they "besist", but it didn't find support.

The main conflict which I deem to be irresolvable is this:
You can argue that logic works the way it does, because it is how our brains work. So logic depends on the mathematical patterns of physics that the universe displays. But on the other hand we are minds, our discussion is based on logic (mostly :wink:), so we can argue that logic needs to be established before we analyze nature, because our minds run on it.

Well let me correct this, it's what my mind runs on, whereas you are just a peculiar physical phenomenon, to talk to on a web forum :tongue2:

It's true that we already have a word for "besist," but that word is "subsist" not "exist" :smile:.

We shall find it convenient only to speak of things existing when they are in time, that is to say, when we can point to some time at which they exist (not excluding the possibility of their existing at all times). Thus thoughts and feelings, minds and physical objects exist. But universals do not exist in this sense; we shall say that they subsist or have being, where 'being' is opposed to 'existence' as being timeless.

http://www.ditext.com/russell/rus9.html

I agree that this distinction can be important. I also agree at least with Bertrand Russell's general ideas on "laws of thought," your idea that our minds run on logic. In The Problems Of Philosophy (http://www.ditext.com/russell/russell.html), chapter 8, Russell criticizes Kant's ideas on mathematics:

Apart from minor grounds on which Kant's philosophy may be criticized, there is one main objection which seems fatal to any attempt to deal with the problem of a priori knowledge by his method. The thing to be accounted for is our certainty that the facts must always conform to logic and arithmetic. To say that logic and arithmetic are contributed by us does not account for this.

He continues with an account of how a priori knowledge is possible and a description of universals and our knowledge of them. I haven't been able to find a more convincing description of the subject, although I'm sure there's been a lot of great criticism in the 100 years since Russell wrote that I am not fully aware of.

Does anyone know of any particularly successful criticism?

[nikman]

Does anyone know of any particularly successful criticism?

Of course you mean favorable criticism, no matter how hedged. Not so easy to come by.

You might try Arthur Pap, "The a Priori in Physical Theory" (from the late 1940s, mentored by Ernst Cassirer). The first important not-particularly-supportive critique may have been Hans Reichenbach's "The Theory of Relativity and A Priori Knowledge" from back in the early 1920s.

Willard Van Orman Quine ... the greatest, but approach this guy with garlic, crucifix and stake, as you're surely aware. He's not your friend.

Here's an interesting recent -- rather technical -- paper which is nothing like what you're after, but still it's certainly challenging. Rub the a priori against Hilbert's 10th problem and you discover you may need to choose between abandoning Kant or accepting God? (Antoine Suarez is a physicist and a longtime auxiliary of Nicolas Gisin's group in Geneva, also the foremost proponent of the "before-before" experiment.)

http://arxiv.org/abs/0809.3691v1

[kote]

Here's an interesting recent -- rather technical -- paper which is nothing like what you're after, but still it's certainly challenging. Rub the a priori against Hilbert's 10th problem and you discover you may need to choose between abandoning Kant or accepting God? (Antoine Suarez is a physicist and a longtime auxiliary of Nicolas Gisin's group in Geneva, also the foremost proponent of the "before-before" experiment.)

http://arxiv.org/abs/0809.3691v1

What's with the implication that I wouldn't be looking for anything technical? I wouldn't consider that paper to be too terribly technical though. I agree with it all the way through the 3rd to last paragraph.

However, the fact that at any time T there is a diophantine equation E such that no human can say whether it is solvable or not, even if the answer to this question exists, proves that mathematical truth will never be completely contained in any human mind. Therefore Mathematics does not exist “a priori” in a human mind.

The paper seems to be assuming a platonic objective existence of mathematical proofs. I'm not sure "math is a priori" supporters believe that the entirety of math is preexisting in our minds. Spinoza was the ultimate rationalist and he didn't think humans could comprehend all rational truths. I think his argument was somewhat similar and can be interpreted as being due to the physical computational limits of human minds. Of course, he also had "God," but that's beside the point.

The argument above also seems very similar to Berkeley's argument for the existence of the real world in the mind of God. Paraphrasing: "There exists a tree even when we aren't looking at it, therefore it must exist in the mind of God," compared to:

Then, if we maintain Kant’s conception that Mathematics is an “a priori” cognition, we are led to conclude that it is “a priori” in some other mind, who is mightier than the human one, a mind who actually contains the whole of Mathematics at once. In this sense one can say that the existence of an omniscient mind (God).

The other option is simply to deny the platonic objective existence of math.

I agree that Quine probably presents the best alternative view. I need to read more of him.

[nikman]

Me, I like the option of denying the platonic objective existence of math. It's clearly implicit (as an option) in the authors' argument despite their theistic thrust.

Of course it's also denying Kant and the a priori. Without that bouncer at the door you can start thinking of math as a symbolized physics, a product of embodied cognition and genetically-archived selected-for patterns of interaction with the physical world going back to whenever life itself began. Then you can proceed and see potential linkage between mathematical proof and physical experiment.

There don't seem to be too many Platonists around these days. A prominent one is Max Tegmark. The physical world is really solidified mathematics, he says; he talks about "mathematical objects." But his corollary goes beyond Kant: whatever can be (coherently) mathematically conceived has no choice but to exist somewhere. Not in this universe, but in another one, somewhere in the infinite multiverse of infinite universes. Where entirely different physical laws, based on different mathematics, hold sway. You kind of wonder how Kant would've responded to that.

[apeiron]

There don't seem to be too many Platonists around these days. A prominent one is Max Tegmark. The physical world is really solidified mathematics, he says; he talks about "mathematical objects." But his corollary goes beyond Kant: whatever can be (coherently) mathematically conceived has no choice but to exist somewhere. Not in this universe, but in another one, somewhere in the infinite multiverse of infinite universes. Where entirely different physical laws, based on different mathematics, hold sway. You kind of wonder how Kant would've responded to that.

Roger Penrose would be another of your true Platonists.

I like your phrasing of the "Tegmarkian" position above as it sharpens the key question.

Do we believe that worlds can exist in an isolated fashion? Or is all reality in interaction?

The idea of infinite variety, a multiverse where all mathematical patterns exist somewhere, demands pretty strong isolation between conflicting worlds.

If instead you are inclined to stressing interaction, process and relationships, then all possible patterns must go through an equilibrating or normalising selective process. The total space of possibility becomes whittled down to the collection of patterning or form that is self-consistence across all worlds. This is the self-organising approach of systems science.

So maths - at some very general level of course - would be homogenous across all actually existent realms (or rather, persistent). Not infinitely heterogenous as Tegmark suggests.

As I say, the philosophical point of focus here becomes the interaction-isolation dichotomy.

Which choice do you believe? That reality is basically a collection of isolated systems (worlds, realms) and that interaction is restricted to the interior of these worlds. Or that reality is basically all in interaction with itself, and isolated systems arise within this landscape by the construction of localised boundaries (in the way a cell creates a relatively isolated chemistry within a membrane).

[nikman]

Roger Penrose would be another of your true Platonists.

Indeed. But Tegmark sure knows how to put on one helluva show. Penrose is a tad, well, donnish.

Do we believe that worlds can exist in an isolated fashion? Or is all reality in interaction?

I believe in relativistic reference frames. I also believe in quantum entanglement. You can have a pair of entangled particles with each particle in a separate relativistic reference frame. The relativistic aspect has no effect on the entanglement. You may not be able to say which particle is measured "first", but the measured correlations don't depend on that. What to conclude? I honestly have no idea.

The idea of infinite variety, a multiverse where all mathematical patterns exist somewhere, demands pretty strong isolation between conflicting worlds.

Tegmark stipulates to that. A few other multiverse theorists have suggested that our universe's dark matter/energy represents leakage from other universes, but that is truly speculative even for that gang.

If instead you are inclined to stressing interaction, process and relationships, then all possible patterns must go through an equilibrating or normalising selective process. The total space of possibility becomes whittled down to the collection of patterning or form that is self-consistence across all worlds. This is the self-organising approach of systems science.

So maths - at some very general level of course - would be homogenous across all actually existent realms (or rather, persistent). Not infinitely heterogenous as Tegmark suggests.

As I say, the philosophical point of focus here becomes the interaction-isolation dichotomy.

Which choice do you believe? That reality is basically a collection of isolated systems (worlds, realms) and that interaction is restricted to the interior of these worlds. Or that reality is basically all in interaction with itself, and isolated systems arise within this landscape by the construction of localised boundaries (in the way a cell creates a relatively isolated chemistry within a membrane).

Why is such a choice necessary? An autopoietic cell would die if it couldn't self-regulate, yet it still imports energy from, and exports waste to, its environment, without whose existence it would also die.

[apeiron]

Why is such a choice necessary? An autopoietic cell would die if it couldn't self-regulate, yet it still imports energy from, and exports waste to, its environment, without whose existence it would also die.

Which was my point. You can get relative isolation, but not complete isolation. And yes, the next step would be to model this more specifically in the language of the second law and dissipative system theory. The interactions have to be aligned along an entropic gradient.

But cells are an example of "interpolated" order - evolved boundaries - and our universe is different in being what I might call an example of "extrapolated" order. Developing boundaries.

The universe "imported" all its energy in one bite at the beginning with the big bang (dark energy of course is a complication to this simple statement). And it is "exporting" all this energy by the creation of a vast heat sink - the expanding void.

So there are two (dichotomously) ways to be a stable dissipative structure. Stay still and transact, or expand and dilute.

The maths of dissipative structure theory would have to be broad enough to capture both kinds of solutions.

[nikman]

But cells are an example of "interpolated" order - evolved boundaries - and our universe is different in being what I might call an example of "extrapolated" order. Developing boundaries.

The universe "imported" all its energy in one bite at the beginning with the big bang (dark energy of course is a complication to this simple statement). And it is "exporting" all this energy by the creation of a vast heat sink - the expanding void.

So there are two (dichotomously) ways to be a stable dissipative structure. Stay still and transact, or expand and dilute.

Can the Universe really be categorized as a dissipative structure, though? Do we know enough about it to make that judgment? Did Prigogine go that far? (Of course I think he balked for a while at classifying life as a dissipative structure.)

And aren't multicellular organisms partly extrapolative, at least in their developmental stages when the same fundamental (epi)genetic complex is involved in the expanding creation of such an enormous variety of cells?

(I may well not be getting something here.)

[apeiron]

Can the Universe really be categorized as a dissipative structure, though? Do we know enough about it to make that judgment? Did Prigogine go that far? (Of course I think he balked for a while at classifying life as a dissipative structure.)


Of course, even applying dissipative structure thinking to bios, life and mind, is still a controversial exercise for many as you say. But not among the theoretical biologists I work with at least.

And extending the idea to the universe itself would be the new rather bold step. There are actually a fair number of journals, conferences and seminars trying to take this tack. But even I say they are 99% flaky.

Yet the universe is clearly dissipating and clearly structured. It is just that we then have to answer the question, well, what is the larger world in which it arose and what exactly is it dissipating to pay for its structuring?

We could answer heat (the big bang planckscale temperature and energy density). Or entropy (the big bang density of microstates - but we have seen how hard it is to make that model comprehensible).

I think there is a more general answer in the idea of vagueness. But that is another story finding little favour (and I would note that Prigogine has a kind of vagueness model for QM if you read End of Certainty, for example).



And aren't multicellular organisms partly extrapolative, at least in their developmental stages when the same fundamental (epi)genetic complex is involved in the expanding creation of such an enormous variety of cells?


Yes, life (and mind) depend on the harnessing of developmental processes. This is a huge issue in biology these days as people try to get past the simplicities of darwinian selection as the primary cause of complexity. It is what Kauffman, Oyama, Salthe and hundreds of others are on about.

Neurogenisis of the infant cortex is a good example of what you say. The free production of neurons and dendrites (simple development) followed by the selection - the constraints - imposed by experience and learning that winnow the pathways.

The term "interpolation" is a bit of jargon from hiearchy theory, stressing the fact that the more complex is nested within the simpler. So it is really a "cross-sectional" view. Life is interpolated as a level of complexity within the physico-chemical realm. And life itself is then an interaction between developmental potentials and evolutionary constraints (the metabolism and repair, or M/R systems, of Rosen).

So I was making the point that cells do have evolved boundaries that create a static context within which new hierarchical levels of development (and evolution) can take place.

The radical idea is then that the universe itself could be read in dissipative structure terms. But you would have to find a different route than the familiar biotic one of interpolation. Playing on words, I suggested extrapolation. Which has the correct sense of free and untrammeled growth or expansion. Whatever was just keeps diverging, keeps happening.

But interpolation is an accepted term and extrapolation would be a non-standard neologism here - yet a nicely dichotomous one I am hoping.

[nikman]

Of course, even applying dissipative structure thinking to bios, life and mind, is still a controversial exercise for many as you say. But not among the theoretical biologists I work with at least.

Are you familiar with the work of Howard Pattee (and more currently Luis Rocha)?

And extending the idea to the universe itself would be the new rather bold step. There are actually a fair number of journals, conferences and seminars trying to take this tack. But even I say they are 99% flaky.

Yet the universe is clearly dissipating and clearly structured. It is just that we then have to answer the question, well, what is the larger world in which it arose and what exactly is it dissipating to pay for its structuring?

I have a problem with the reference in your previous post to the expanding void as a vast heat sink. In order to have a heat sink you need matter (per the Second Law ... heat transfer is from hotter to colder molecules). Are you casting dark matter in that role? Unless matter is continually created along with the expansion (per Fred Hoyle's theory) why would you need expansion in order to have a heat sink?


I think there is a more general answer in the idea of vagueness. But that is another story finding little favour (and I would note that Prigogine has a kind of vagueness model for QM if you read End of Certainty, for example).

I haven't read it but I think I should. I didn't know he'd discussed QM.

Yes, life (and mind) depend on the harnessing of developmental processes. This is a huge issue in biology these days as people try to get past the simplicities of darwinian selection as the primary cause of complexity. It is what Kauffman, Oyama, Salthe and hundreds of others are on about.

By "simplicities of Darwinian selection" are you referring to the "radical adaptationists" (like Dawkins and his large Mini-Me, Dennett)? That issue's also being addressed by Allen Orr, among others. Also, on a more purely philosophical level, Jerry Fodor.

The radical idea is then that the universe itself could be read in dissipative structure terms. But you would have to find a different route than the familiar biotic one of interpolation. Playing on words, I suggested extrapolation. Which has the correct sense of free and untrammeled growth or expansion. Whatever was just keeps diverging, keeps happening.

But interpolation is an accepted term and extrapolation would be a non-standard neologism here - yet a nicely dichotomous one I am hoping.

I believe I need to know more than I do about hierarchy theory to follow this line of thought ....

[apeiron]

Hi Nikman

Are you familiar with the work of Howard Pattee (and more currently Luis Rocha)?


Yes, Pattee is one of the key thinkers I have worked with. My position is not identical with his by any means, but it certainly grew out of debates we had about the epistemic cut and the dichotomy involved. He is of course a hierarchy theorist and close colleague of Robert Rosen and Stan Salthe. And these three would be the best I have come across.

I've read Rocha's work and I think he was on some of the old chat forums like VCU Complexity, but felt like many of these guys' grad students, the original deeper ideas have become rather diluted in repetition - homogenised to fit with the mainstream to some extent.

Is there some particular aspect of Rocha you are thinking of here?


I have a problem with the reference in your previous post to the expanding void as a vast heat sink. In order to have a heat sink you need matter (per the Second Law ... heat transfer is from hotter to colder molecules). Are you casting dark matter in that role? Unless matter is continually created along with the expansion (per Fred Hoyle's theory) why would you need expansion in order to have a heat sink?


Here I would cite Lineweaver's MEP papers as a useful source.

The basic story of the universe is about the cooling of radiation via expansion - redshifting. Lineweaver runs the figures. The dissipation of matter becomes almost an afterthought. A secondary "interpolated" level of dissipative structure in fact.

So radiant matter does get "sunk" by simple expansion. And if protons, and whatever dark matter is, are subject to decay (recycled through black holes eventually if need be), then they will join this story.


By "simplicities of Darwinian selection" are you referring to the "radical adaptationists" (like Dawkins and his large Mini-Me, Dennett)? That issue's also being addressed by Allen Orr, among others. Also, on a more purely philosophical level, Jerry Fodor.


Dennett :tongue2: Nice guy but shallow as....

And as for the current Darwin backlash, I'm not really following it closely as it is not a critical issue for me at the moment. Also it is a rather sociological movement, the science at risk of being co-opted by the intelligent design crew.

I've worked closely with Stan Salthe for many years and so I've heard much of it before....

http://cache.zoominfo.com/CachedPage/?archive_id=0&page_id=-1816042126&page_url=%2f%2fwww.scoop.co.nz%2fstories%2fHL0807% 2fS00053.htm&page_last_updated=7%2f6%2f2008+2%3a17%3a52+PM&firstName=Stanley&lastName=Salthe


I believe I need to know more than I do about hierarchy theory to follow this line of thought ....

If you are really interested, then PM me and I can send you links. The fundamentals of hierarchy theory are what I am researching.

[nikman]

Is there some particular aspect of Rocha you are thinking of here?

He worries a lot about the epistemic cut, and (maybe you consider this compromising with the mainstream) building bridges between dynamicists and computationalists. But I don't know of anyone else who's been doing more to carry on the Pattee tradition. (I'm not especially familiar with HP's work as a hierarchy theorist, which I know he was back in the 1970s. He caught my attention in his broader role as a theoretical biologist.)

Here's some Rocha stuff I know about and like, some or all of which you may very well know too:


"Material Representations: From the Genetic Code to the Evolution of Cellular Automata" (with Wim Hordijk)

http://informatics.indiana.edu/rocha/ps/caalife04.pdf


"Artificial Semantically Closed Objects"

http://informatics.indiana.edu/rocha/ps/tilsccai.pdf


"Eigenbehavior and Symbols"

http://www.informatics.indiana.edu/rocha/ps/sr.pdf


You may very well also know this paper by Evan Thompson, but anyway I like to toss it into the mix whenever I see an opening. He worked with Dennett back in the early 90s, it's true, but he's come a long way since then:

"Symbol Grounding: A Bridge from Artificial Life to Artificial Intelligence"

http://individual.utoronto.ca/evant/SymbolGrounding.pdf


I'm very much opposed to functionalism and the whole substrate neutrality/multiple realizability ethos, but I also believe there has to be a natural limit to reductionism. I see a possible solution emerging from the QM informatics work by Zeilinger, Brukner et al and abetted by onlookers like Hans C von Baeyer. I feel an affinity toward the basic ideas behind biosemiotics (including the work of Claus Emmeche and Jesper Hoffmeyer) but at the moment it's somewhat, well, hand-wavy.

I'm in a rush much of the time right now, but will get back.

[apeiron]

(maybe you consider this compromising with the mainstream) building bridges between dynamicists and computationalists.

Yes. Simply put, I was finding that many wanted to "find the truth" of a dynamic view by reframing it in computational terms. But then I came away from debates with Pattee, Salthe and others wanting to do the opposite - to find a dynamic way to frame the truths of computationalism. Define an organic logic to complement the more familiar mechanical logic, so to speak.

Other grad students of Pattee, like Peter Cariani, also had more radical ambitions and consequently, I would say, struggled to make it in academia.

[nikman]

Frankly, I see no important difference between Rocha and Cariani ...

[nikman]

Just re-read Cariani's "Symbols and Dynamics in the Brain" (linked from Rocha's website) and Rocha's papers linked above ...

Are you perhaps factoring in some personal dimension here?

[SixNein]

We might, indeed at first suppose that the proposition 7 + 5 = 12 is a merely analytical proposition, following (according to the principle of contradiction) from the conception of a sum of seven and five. But if we regard it more narrowly, we find that our conception of the sum of seven and five contains nothing more than the uniting of both sums into one, whereby it cannot at all be cogitated what this single number is which embraces both. The conception of twelve is by no means obtained by merely cogitating the union of seven and five; and we may analyse our conception of such a possible sum as long as we will, still we shall never discover in it the notion of twelve. We must go beyond these conceptions, and have recourse to an intuition which corresponds to one of the two− our five fingers, for example...Do you agree with Kant here? Do you agree that our conception of the number 12 depends to some degree on experience and isn't something we can derive rationally?
/QUOTE]

I disagree with Kant.

All you need is rules.

[JoeDawg]

All you need is rules.

But where does one get the rules?

Without some foundation, the selection process is random.

[SixNein]

But where does one get the rules?

Without some foundation, the selection process is random.

The foundation is the rules. We could invent our own rules and create our own numbers.

The letter @ is a number.
Every number has a subsequent number.
No numbers share the same former number.
@ does not come after any number.
Any property that belongs to the number @, and any property that belongs to subsequent numbers, belongs to all numbers.

Using the above rules, I have created the following list:
@, A, B, C, D, E, F, G, ... N

But you need more rules to do anything with it.

In a basic nutshell, the rules of the number serve as the foundation.

[Sorry!]

The foundation is the rules. We could invent our own rules and create our own numbers.

The letter @ is a number.
Every number has a subsequent number.
No numbers share the same former number.
@ does not come after any number.
Any property that belongs to the number @, and any property that belongs to subsequent numbers, belongs to all numbers.

Using the above rules, I have created the following list:
@, A, B, C, D, E, F, G, ... N

But you need more rules to do anything with it.

In a basic nutshell, the rules of the number serve as the foundation.

The question was WHERE do you get the rules, not how can we formulate them. So explain why you have chosen those rules above... and what parameters we should use to select further rules?

[SixNein]

The question was WHERE do you get the rules, not how can we formulate them. So explain why you have chosen those rules above... and what parameters we should use to select further rules?

You obtain the rules from thinking and asking questions. You could create different rules that are weaker or stronger, and the nature of your numbers would change depending upon the rules. In this example, I just spit out a rehash of Peano's axioms.

[SixNein]

Most?

Most is correct.

The continuum hypothesis is independent of our current system, so we cannot solve it without creating new axioms. Quite a few different problems are independent.

[JoeDawg]

You obtain the rules from thinking and asking questions. You could create different rules that are weaker or stronger, and the nature of your numbers would change depending upon the rules. In this example, I just spit out a rehash of Peano's axioms.

How do you measure strength of a rule?

[SixNein]

How do you measure strength of a rule?

Through the means of consistency.

I personally think Kant is off base. Counting is one human field that can be found in nature. A wolf can count, and it doesn't use its fingers. The concept of numbers is extremely primitive even though the definition of numbers is quite complex. Although our counting abilities are developed through external means, life could find the means to count through thought.

[JoeDawg]

Through the means of consistency.


Consistency demands that you have at least 2 axioms, how do you select the first one?
This is where experience comes in. Even the very idea of 'consistency' comes from experience. We value consistency because of its value in predicting outcomes which are beneficial to us. We evolved this ability, because its benefitial to survival.

Although our counting abilities are developed through external means, life could find the means to count through thought.

And a millions monkeys typing on keyboards for an infinite amount of time could write the complete works of shakespeare. But 99.9999...% of what they churn out would be nonesense, and some would be 'consistent' but not resemble reality.

Math was invented by generalizing experience.

[SixNein]

Consistency demands that you have at least 2 axioms, how do you select the first one?
This is where experience comes in. Even the very idea of 'consistency' comes from experience. We value consistency because of its value in predicting outcomes which are beneficial to us. We evolved this ability, because its benefitial to survival.



And a millions monkeys typing on keyboards for an infinite amount of time could write the complete works of shakespeare. But 99.9999...% of what they churn out would be nonesense, and some would be 'consistent' but not resemble reality.

Math was invented by generalizing experience.

Thought is an experience. Mathematics is a manor of symbolic thinking.


When was math invented exactly? Mathematical abilities can be found in other species.

[JoeDawg]

Thought is an experience. Mathematics is a manor of symbolic thinking.

Thought is qualitatively different from sense experience.

When was math invented exactly? Mathematical abilities can be found in other species.
Hard to say, but geometry goes back at least as far as the Ancient Egyptians. Before writing, I'm not sure one could even have a formal system.

Mathematical ability derives from logical ability, which comes from observing the world.
If you've ever watched a baby learn, you'll see how they start to form logical patterns.

Formalized mathematics is different from ability however. Simple counting doesn't really require a formal system. Few, some, many.... is counting.

A formal system of math probably requires writing.

[SixNein]

Thought is qualitatively different from sense experience.

Hard to say, but geometry goes back at least as far as the Ancient Egyptians. Before writing, I'm not sure one could even have a formal system.

Mathematical ability derives from logical ability, which comes from observing the world.
If you've ever watched a baby learn, you'll see how they start to form logical patterns.

Formalized mathematics is different from ability however. Simple counting doesn't really require a formal system. Few, some, many.... is counting.

A formal system of math probably requires writing.

In the particular situation we are discussing, the person probably can't do much math. In fact, a person isolated from the rest of humanity could not do much either. But I believe the person could grasp the concept of numbers without senses. Could you tell if you had more than one thought?

[JoeDawg]

Could you tell if you had more than one thought?

I think this is one of the problems faced by those who want to create an AI. I think our consciousness is a function of interacting experiences from multiple stimuli. Creating a brain in box, or brain in a vat, means there is really only one source of stimuli. If we want to mimic intelligence, I think we need to include a variety of input sources. A robot with eyes, ears... a sense of touch....etc..

But to answer your question, I think you would have to have a need to distinguish between thoughts, and that need would undoubtedly rely on an external influence.

[wofsy]

A pretty big problem from what I have read. I think you're idealizing mathematics, like the ancient greeks did with geometry. It all happens in the human brain. And different cultures have developed different mathematical systems. Ours has simply absorbed all the aspects we find most useful.

Math is in large part analytic, but its axioms and basic logic are derived from experience. All logic comes from how we see the world working. Math is just a way to represent and predict experience using highly abstract language.

Axioms are little more than assumptions, or constraints. And those constraints are based on our experiences in the world.

A couple unrelated thoughts: They are meant as talking points not as assertions of anything that I believe.

- in some sense everything happens in the human brain. Mathematics and experience are inescapably unified.

- Logic is intrinsic. If it were not we would not be able to think.

- "Derived from expeience" is a vague idea. Derived has no rigorous definition.

- Math is not "just" a way to represent and predict experience". Here is a quote from Felix Klein's introduction to his book, Riemann's Theory of Algebraic Functions and their Integrals."

"He(Riemann ) had in mind far more general methods of determination than those we employ in the following pages;methods of determination in which physical analogy ...fails us."

- Experience suggests an underlying mathematical reality but that reality supercedes direct experience

[apeiron]

A couple unrelated thoughts: They are meant as talking points not as assertions of anything that I believe.

- in some sense everything happens in the human brain. Mathematics and experience are inescapably unified.

- Logic is intrinsic. If it were not we would not be able to think.

- "Derived from expeience" is a vague idea. Derived has no rigorous definition.

- Math is not "just" a way to represent and predict experience". Here is a quote from Felix Klein's introduction to his book, Riemann's Theory of Algebraic Functions and their Integrals."

"He(Riemann ) had in mind far more general methods of determination than those we employ in the following pages;methods of determination in which physical analogy ...fails us."

- Experience suggests an underlying mathematical reality but that reality supercedes direct experience

If you want to demystify and make rigorous the relationship between minds and worlds, Rosen's modelling relations work serves as a good template.....

http://www.panmere.com/?page_id=18

http://www.osti.gov/bridge/servlets/purl/10460-5uGkyu/webviewable/10460.pdf