Clayton: Zangelbert Bingledack:0 is a bookkeeping symbol, Yes, that is, in fact, how the use of zero originated but I don't think it helps your point in any way. I asked whether zero, in fact, exists and your response is "it's a book-keeping symbol."
Zangelbert Bingledack:0 is a bookkeeping symbol,
Yes, that is, in fact, how the use of zero originated but I don't think it helps your point in any way. I asked whether zero, in fact, exists and your response is "it's a book-keeping symbol."
actually i said, "0 is a bookkeeping symbol, which doesn't represent anything all by itself". if a symbol doesn't represent anything, there is nothing subject to a question of existence or non-existence, or even the possibility of impossibility of any action.
the only interpretation left is to ask whether the symbol itself, a round shape, "exists". and yes, that shape does exist, as an ink mark on a piece of paper, a shape envisioned1, or even a 3D object...although an ink mark is also 3D of course. but i'll settle this once and for all with photographic proof!
[a zero made of mylar]
1depending on the definition of "exist"...which is THE underlying issue we would have to resolve for all of us to be happy campers in this thread, i think.
Clayton:Note also that you conveniently ignored the imaginary unit, i. Is it a book-keeping symbol? If so, to what does i correspond?
just like 1, the so-called imaginary unit i is a symbol that corresponds to whatever aspect of experience we link it with. since i'm not familiar with any of the fields where i is used and did not have an easy example to work with, i left it out as i thought the point had already been made. anyway, the principle is the same:
if i decide "1" means "1 apple", then "1 + 1 = 2" is a class of praxeological statement-forms telling me that, for instance, if i find one apple and find another apple, I have found two apples.
i am no electrical engineer, but suppose we decide that the imaginary unit i corresponds to something in electronic circuitry where we have linked "-1" with a certain object (or physical state of affairs; abbreviated hereinafter). suppose we have also defined what multiplication corresponds to in that system. then perhaps we find an object that can be multiplied by itself to yield that "-1" object. we name it i. why? because then mathematics has a nice set of praxeological tautologies that can tell us useful things about that system that we would have a hard time deducing otherwise.
SUMMARY: what is i? a symbol that corresponds with whatever we decide it corresponds to for a given application, of course after we have already decided what 1, -1, 0, and so on correspond to. once we have done that, we can make use of the many mathematical theorems about complex numbers, with the faith that the mathematicians have done the hard work of logical proof for us already.2 unlike 1, i has no readily familiar example of correspondence to most people's everyday experience, such as counting how many noses you have on your head. again, the symbol does "exist" but only in the sense of being a blot of ink on a piece of paper, in image held in the mind, and so on. some are shown below...
and here's that in 3D!
[i made of rubber]
2with the all-important caveat that we have to make sure the mathematical symbols are linked with correspondents in actual experience in such a way that the formal manipulation of mathematical symbols cannot end up telling us nonsense.
now this line of discussion has gone like this...
and finally i would like to say that one could, by the same logic outlined in this post, decide something that "infinity" corresponds to in actual experience. but insofar as "infinity" can be usefully linked with experience, it cannot mean what it does in most math works. for example, we decide "i have infinity apples" shall mean "i have more apples than i could ever need or want or whatever".
in short, we cannot "mean" what we cannot think or imagine, as i understand the verb "to mean": to hold a thought in one's mind and utter a sound (or write symbols) in an attempt to get the listener to hold the same thought in her mind.
Clayton:Mathematical formalism is any discipline of symbol manipulation. What the symbols represent (your primary concern) is irrelevant to the study of the rules of manipulation. The symbols could represent nothing that is part of our experience or even anything that we can envision or imagine in our minds, even in principle. This is praxeologically possible for the same reason that any aesthetic endeavor is... for the sheer pleasure of watching the symbols be manipulated.
well Clayton, i fully agree, and see-and-say is a good illustration of that.
and i know this thread has been long and chaotic, and probably feels quite weird.
BUT, the original reason we were talking about mathematical formalism was that i was saying the notion of infinity was brought into other fields, which i just remembered includes religion!, as if it was valid because those ever-rigorous math people all agree on it.
in short, if we agree that infinity is a symbolic formalism, i suspect there is no disagreement in this neighborhood of the thread.
baxter:>"0 is a bookkeeping symbol, which doesn't represent anything all by itself." The view of it as a formal symbol on a piece of paper is only one viewpoint. Zero is also a concept grasped by the human mind. It's the number of items left when you start with one item and take one away (number-as-a-construction). It's the number of steps you have to take to stay in the same place (number-as a-motion metaphor). It's where you're standing right now (number-as-a-place metaphor).
The view of it as a formal symbol on a piece of paper is only one viewpoint. Zero is also a concept grasped by the human mind. It's the number of items left when you start with one item and take one away (number-as-a-construction). It's the number of steps you have to take to stay in the same place (number-as a-motion metaphor). It's where you're standing right now (number-as-a-place metaphor).
i essentially agree with this: see the bold, or see the post right before this one for a more complete answer.
baxter:>"1 - 1 = 0" is a praxeological statement-form I don't see what it has to do with praxeology. It is simply a datum. Even birds understand 1 - 1 = 0. When one threat comes near its nest, and then one threat is seen to be leaving, the bird will relax.
I don't see what it has to do with praxeology. It is simply a datum. Even birds understand 1 - 1 = 0. When one threat comes near its nest, and then one threat is seen to be leaving, the bird will relax.
that's actually a good example. i mean praxeology not strictly as "human action" but action by any entity capable of rational decision-making, which i think [personal opinion] probably includes birds, dogs, cats, and many other animals. (i can't recall if Mises agrees with this or not.)
baxter:>most of mathematics is best demarcated as a sub-branch of praxeology Are you saying that Earth, Venus, and Mercury weren't 3 planets, and didn't travel in ellipses, in the aeons when Earth was devoid of people?
Are you saying that Earth, Venus, and Mercury weren't 3 planets, and didn't travel in ellipses, in the aeons when Earth was devoid of people?
i said, "it is a question of how one defines mathematics, but i would argue that at least most of mathematics is best demarcated as a sub-branch of praxeology". if we define the purview of mathematical statements to include statements that speak directly about the motion of physical objects in an elliptical path, rather than about ellipses as ideal shapes, it would not then be fully contained in praxeology. it may, however, be fully contained within "praxeology together with thymology", except i am not prepared to claim that because i am not sure the exact definition of thymology.
baxter:The mind has direct access to 3D geometry through the senses of touch and proprioception.
that may actually be good clue. understanding1 3D objects is probably related to how we sense our own bodies, first and foremost.
1which just means, "figuring out what a given object of perception means for the pleasantness or unpleasantness of one's future experience"
Jeremiah Dyke: 2D creatures would only see lines
2D creatures would only see lines
can you actually imagine this? i know you can imagine flat creatures with the thickness of a piece of paper living in a box that is only as thick as a piece of paper, and only seeing long, thin rectangles...but can you really imagine 2D? can you imagine something of zero thickness? and if you cannot, how can you actually mean anything by the words "2D creature"?1
1assuming you agree with this definition: mean (v.): to hold a thought in one's mind and utter a sound (or write symbols) in an attempt to get the listener to hold the same thought in her mind. if you don't agree with the definition, why not?
Zangelbert Bingledack:
"I mean praxeology not strictly as "human action" but action by any entity capable of rational decision-making, which i think [personal opinion] probably includes birds, dogs, cats, and many other animals. (i can't recall if Mises agrees with this or not.)"
Mises:
"The a priori sciences---logic, mathematics, and praxeology---aim at knowledge unconditionally valid for all beings endowed with the logical structure of the human mind." (Human Action)
Basically the same idea.
*****
"I would argue that at least most of mathematics is best demarcated as a sub-branch of praxeology".
Great point.
Would this be for the reason that we can conceive mathematics as a science instructing on the necessary (a priori or apodictic) consequences of a specific group or kind of human acts?
E.g., If you have 4 (your current state of affairs), and you take 2 away (an action or a means, depending on how one conceives it), you will then have 2 (the necessary end or result of your action). ??
Mathematics can be demarcated or conceived as a sub-branch of praxeology for a reason such as this ?
Regarding the above two ideas; that praxeology is a science describing the laws that apply to acting beings generally (human or not), and that mathematics can be conceived as a branch of praxeology, may I ask, through what course of study were you able to reason to these conclusions ?
Adam
"It would be preposterous to assert apodictically that science will never succeed in developing a praxeological aprioristic doctrine of political organization..." (Mises, UF, p.98)
Mathematics have little to nothing to do with praxeology, which itself to me is a little suspect.
And all accepted mathematics is rigorous, it would not be accepted otherwise.
Yes, I think "symbolic formalism" would be a sufficient description of infinity.
I think the issue merits further thought, however. In this lecture, Steven Pinker argues that human language actually contains a latent theory of physics within it and he delves into grammatical constructs to illustrate this. He points out that we say that an ant crawls along the edge of a plate but across the surface of the plate. You do not say "the ant walked along the plate" if you're trying to communicate that the ant walked across the surface of the plate. The mind is dividing the plate into idealized 1-dimensional and 2-dimensional surfaces and rejects the usage of the word along when referring to the act of crossing a 2-dimensional surface. You do not say "he walked along the floor" you say "he walked across the floor."
I think that much of mathematics actually resides in this realm of shared idealizations of the physical world. Many of the things that are taken to be "natural" in mathematics - for example, prime numbers - are actually very strange when you think about it and why we should find those particular kind of abstract objects so interesting is puzzling and, I think, deserves an explanation and I think Pinker's theory of language goes a long way to doing just this.
I believe that infinity is just exactly one of these kind of shared idealizations.
Clayton -
>Mathematics have little to nothing to do with praxeology
Yes. Praxeology involves axioms that are either a priori or empirically true. It is not always so with mathematics: there, one is free to choose crazy axioms and then study the consequences. The results may or may not clearly correspond with anything in the real world. A striking example is the plethora of superstring theories, which amount to little more than playing mathematical games.
>And all accepted mathematics is rigorous
This is a contentious issue. "Accepted" is contingent on which axioms are adopted. Many of Euler's results (e.g. the reflection formula for the Zeta function) are not rigorous by modern standards, but they may be true and accepted before "rigorous" evidence is contrived after the fact. Computer-generated proofs may be rigorous, yet so long and complicated, that they are not accepted. The Riemann hypothesis is seemingly accepted, judging by its frequent use in proofs, despite not being proven itself.
baxter: Yes. Praxeology involves axioms that are either a priori or empirically true. It is not always so with mathematics: there, one is free to choose crazy axioms and then study the consequences. The results may or may not clearly correspond with anything in the real world. A striking example is the plethora of superstring theories, which amount to little more than playing mathematical games.
Just because every praxeologist stuck to applied praxeology doesn't mean that pure praxeology isn't possible.
Ludwig von Mises: The scope of praxeology is the explication of the category of human action. All that is needed for the deduction of all praxeological theorems is knowledge of the essence of human action. It is a knowledge that is our own because we are men; no being of human descent that pathological conditions have not reduced to a merely vegetative existence lacks it. No special experience is needed in order to comprehend these theorems, and no experience, however rich, could disclose them to a being who did not know a priori what human action is. The only way to a cognition of these theorems is logical analysis of our inherent knowledge of the category of action. We must bethink ourselves and reflect upon the structure of human action. Like logic and mathematics, praxeological knowledge is in us; it does not come from without. All the concepts and theorems of praxeology are implied in the category of human action. The first task is to extract and to deduce them, to expound their implications and to define the universal conditions of acting as such. Having shown what conditions are required by any action, one must go further and define--of course, in a categorial and formal sense--the less general conditions required for special modes of acting. It would be possible to deal with this second task by delineating all thinkable conditions and deducing from them all inferences logically permissible. Such an all-comprehensive system would provide a theory referring not only to human action as it is under the conditions and circumstances given in the real world in which man lives and acts. It would deal no less with hypothetical acting such as would take place under the unrealizable conditions of imaginary worlds. But the end of science is to know reality. It is not mental gymnastics or a logical pastime. Therefore praxeology restricts its inquiries to the study of acting under those conditions and presuppositions which are given in reality. It studies acting under unrealized and unrealizable conditions only from two points of view. It deals with states of affairs which, although not real in the present and past world, could possibly become real at some future date. And it examines unreal and unrealizable conditions if such an inquiry is needed for a satisfactory grasp of what is going on under the conditions present in reality.
The scope of praxeology is the explication of the category of human action. All that is needed for the deduction of all praxeological theorems is knowledge of the essence of human action. It is a knowledge that is our own because we are men; no being of human descent that pathological conditions have not reduced to a merely vegetative existence lacks it. No special experience is needed in order to comprehend these theorems, and no experience, however rich, could disclose them to a being who did not know a priori what human action is. The only way to a cognition of these theorems is logical analysis of our inherent knowledge of the category of action. We must bethink ourselves and reflect upon the structure of human action. Like logic and mathematics, praxeological knowledge is in us; it does not come from without.
All the concepts and theorems of praxeology are implied in the category of human action. The first task is to extract and to deduce them, to expound their implications and to define the universal conditions of acting as such. Having shown what conditions are required by any action, one must go further and define--of course, in a categorial and formal sense--the less general conditions required for special modes of acting. It would be possible to deal with this second task by delineating all thinkable conditions and deducing from them all inferences logically permissible. Such an all-comprehensive system would provide a theory referring not only to human action as it is under the conditions and circumstances given in the real world in which man lives and acts. It would deal no less with hypothetical acting such as would take place under the unrealizable conditions of imaginary worlds.
But the end of science is to know reality. It is not mental gymnastics or a logical pastime. Therefore praxeology restricts its inquiries to the study of acting under those conditions and presuppositions which are given in reality. It studies acting under unrealized and unrealizable conditions only from two points of view. It deals with states of affairs which, although not real in the present and past world, could possibly become real at some future date. And it examines unreal and unrealizable conditions if such an inquiry is needed for a satisfactory grasp of what is going on under the conditions present in reality.
It would be more accurate to say that his end was to know reality, so, in his praxeology, he stuck to using "realistic axioms".
If I wrote it more than a few weeks ago, I probably hate it by now.
>if we are not talking about bending in any direction, in what sense are we saying that a long, thin pole "bends"?
A pole in a gravitational field bends in the same way that a geodesic on the Earth bends. A flat-earther walking along the geodesic might not perceive the bend, but a large-scale geometrical study would reveal it. The numerical magnitude of bending - the curvature - can be computed from the Ricci or metric tensor.
>i found the following definition in Nonstandard Analysis, by Dr. J. Ponstein...this definition smuggles in the idea that it is even possible or meaningful
Indeed, the possibility and meaning is imbued in the definitions. Just like defining the successor operation and saying 0'=1 assumes that its even possible or meaningful to have numbers other than zero. Your insistence on "existence proofs" makes you sound like a constructivist-type mathematician http://en.wikipedia.org/wiki/Constructivism_(mathematics). I'm not sure how far that mentality can be carried, since presumably they depend on axioms as well.
Clayton: In the same sense that Hume is speaking of an actual point. He notes that if you used a device you would be able to separate the dot again, even after you have stepped back from it. By backing up, you accept that you are not altering the point itself, only your view of the point. Hume is simply noting that our perception of things is what it is and when you step back from the point, a condition arises where the point itself is indivisible by your mind in its own perceptual space. This is not a statement that the physical world itself has changed simply because you backed up, which is what he means when he says that you could use a telescope to separate the light beams and see the dot in more detail even after having backed away from it. If you back away from a series of closely spaced (yet separated) dots, they will start to look like a solid line. But if you got the telescope out, you would again be able to see that they are, in fact, separated points. The points do not physically merge together by virtue of your stepping back any more than the point on the paper becomes physically indivisible by virtue of your stepping back.
In the same sense that Hume is speaking of an actual point. He notes that if you used a device you would be able to separate the dot again, even after you have stepped back from it. By backing up, you accept that you are not altering the point itself, only your view of the point. Hume is simply noting that our perception of things is what it is and when you step back from the point, a condition arises where the point itself is indivisible by your mind in its own perceptual space. This is not a statement that the physical world itself has changed simply because you backed up, which is what he means when he says that you could use a telescope to separate the light beams and see the dot in more detail even after having backed away from it.
If you back away from a series of closely spaced (yet separated) dots, they will start to look like a solid line. But if you got the telescope out, you would again be able to see that they are, in fact, separated points. The points do not physically merge together by virtue of your stepping back any more than the point on the paper becomes physically indivisible by virtue of your stepping back.
Just thought of something.
David Hume was aware of the technology used to spread out the "points" making up space, such as telescopes and microscopes, but he wasn't aware of the technology used to spread out the time including those points, such as slow-motion cameras. Probably you won't agree with my exposition of that, but try to bear with me real fast. There are situations where the frame-rate of our vision isn't high enough to capture something. For example, just like the fact that you could back up from something until your mind can't divide it anymore, you could launch something past your field of vision so fast that your mind can't capture it at all. But that doesn't mean that it isn't there: It would be possible to use special equipment, such as a slow-motion camera, to "spread out" the "time frames", so you could see it. But that's exactly analogous to what he was talking about when he was saying that just because we can't see something doesn't mean that it isn't there, because you could use a small telescope to "spread out" the light that was always flowing from the dot that he was talking about, so you could start seeing it again. Just like saying that something moving too fast to see it doesn't mean that it doesn't even exist, saying that you're too far away from something to see it doesn't mean that it isn't there. Just like the fact that something is moving so fast that we can't see it anymore doesn't destroy it, the fact that something is so far away that we can't see it anymore doesn't destroy it! Just like we can't launch something so fast as to destroy the matter making it up, we can't step back far enough as to destroy the point, or whatever we're talking about.
So what was it that I was trying to explain?
Well, first, let's look out for a possible equivocation on the word "point".
A point could either be a 3D object in space - which you called an "actual point" - or it could be a 2D point making up a part of your field of vision - which you called a point "in your mind's own perceptual space". Let me refer to an actual point - a 3D dot - as a "dot", and refer to a point in your mind's own perceptual space - a 2D point making up a piece of your field of vision - as a "colored point". Maybe this will help clear up the confusion. Let's take David Hume's example in those terms. Draw a dot on your wall, and step away from it until it disappears. Well, right before it disappears, your mind is representing it as a single colored point on your visual field. I'm not saying that the dot - which is a real existence - is the colored point on your visual field. I'm just saying that, right before it disappears, your mind is representing it as a single colored point on your visual field. It's only one view of the dot, and it happens to be a view where your mind is showing it to you as just a single colored point. The import of this example is simply that it allows you to see how small a single "pixel" of your visual field is. It shows you how small a piece of color on your visual field can get before it disappears.
But why did I care to bring up that distinction?
I think that a lot of the confusion here probably came from the fact that we were equivocating on the word "point". I was saying that we see an arrangement of 2D points for each moment of time that we experience, and you thought that I meant that "real time" - whatever that is - actually has moments, and that they are made up of 2D points. Well, that would be totally absurd, and it isn't what I'm saying. I'm saying that our experience of time (time in "your mind's own perceptual space") is broken up into a series of moments, and that each experience (a single view of space "in your mind's own perceptual space") is broken up into an arrangement of 2D colored points. I'm not sure that you will accept this yet, but maybe this has cleared up some of the confusion. Either way, I'm going to move on, and see where else I can go in this post.
We might only be able to experience a certain frame-rate, but things like slow-motion cameras show us that things are going on that our frame-rate can't pick up. It shows us that there's more to what we're seeing right now than just the X number of frames that we see in any given span of time. We find that we can "spread" a span of time out, so we can more things about it. We notice that we can spread an event taking up 60 frames into an event taking up 600 frames, and find 10 times the amount of information. (Notice that high frame-rate cameras are useless unless we slow down the speed! If there are 600 frames to be seen per second, and we can only see 60 per second, we would need to "spread it out" from 1 second of time into 10 seconds of footage.) It isn't clear - at least to me - how far we could go with this process, but who knows. We used to only be able to magnify things up to whatever amount of times, and now we can do it a lot more. I'm sure that telescopes were a lot weaker back when David Hume was around, but we might still be able to go farther. In the same vein, I'm sure that slow-motion was a lot worse 20 years ago, but who knows how far we will be able to go. In this sense, both "real space" and "real time" might be "infinitely divisible" - divisible to the point that any loss of information is totally negligible, but that doesn't refute what I was saying, that each of our "views" of this "real space" or "real time" are finitely divisible and made up of simple, indivisible parts. It's extremely important to understand this distinction - that between "real space/time" and our "views" of it.
But, indeed, let's not get too carried away with this: It would be a backwards method to say that we go from "real space/time" to our "views" of it. It's really that sets of our views - our views organized in certain ways as to satisfy our desires - are what define this "real space and time". Real space and time is nothing but sets of our views of them. Even calling them "our views of them" is an artifact of the backward method. Really it's just this: We experience things, and then we group them into categories as to reflect how we feel about them in terms of satisfying our desires - of course it all comes back to human action.
In short: Space and time might be infinitely divisible - if that means being divisible until we get bored, but each member of the categories making these ideas up aren't.
Let me go out on a limb here and start talking about something that I don't know anything about.
Really I don't know anything about calculus, but based on random things that I have heard about it over the years, I have always got the feeling that calculus is an attempt to model things that are inherently "continuous" - that is, we can divide them into we don't find any utility in continuing - in our mind's framework, which is "discrete". It's something like this. We divide our view of motion into as many parts as we need until we think that the calculations are coming out well, but our views of motion are always a sequence of discrete parts. And doesn't the fact that dividing our views of something into so many parts gives us better answers say something about "real time", like that it is "continuous" in a sense (really just in the sense that accurately modeling physics requires us to model motions as many more parts than we could experience in normal time)? Maybe you could set me straight here, because I'm working with a bunch of wild assumptions, and no better understanding of calculus than a class that I failed in high school and a few Wikipedia articles that I read three years ago.
Now let me get back to something that I actually know something about.
Hopefully you got something out of this post, and maybe next I will try to explain how we move from our view of time - a sequence of appearances made up of colored points arranged in 2 dimensions - to our idea of real space and time. Maybe in the next post I will try to give a quick overview of how we come up with our idea of 3D objects when we don't have anything to work with but a sequence of 2D images! Something fascinating about it is that it's possible to conceive of a situation - an arrangement of desires - that would make 3D objects not make any sense. 3D objects aren't an ultimate given to our mind - like causality, but are a contingent fact of our world!
Adam Knott:"I would argue that at least most of mathematics is best demarcated as a sub-branch of praxeology". Great point. Would this be for the reason that we can conceive mathematics as a science instructing on the necessary (a priori or apodictic) consequences of a specific group or kind of human acts? E.g., If you have 4 (your current state of affairs), and you take 2 away (an action or a means, depending on how one conceives it), you will then have 2 (the necessary end or result of your action). ?? Mathematics can be demarcated or conceived as a sub-branch of praxeology for a reason such as this ?
yah that'd be it.
is not that the whole sales pitch of math to non-mathematicians? "INSTRUCTION SETS SOLD HERE! get your instruction sets, then all ya gotta do figure out how they link up with whatever real-world issue you're facing. after that, the math geniuses have done all the hard logical figuring work for you. just plug n' chug!"
but i actually will give a counterexample that seems to go against this in order to make the point:
<9x9=81 could be used to learn something completely unrelated to human action: i know there are 9 boxes of 9 donuts each in my kitchen, so i can know there are at least 81 donuts there. "there are 9 donuts" is not an action, so although 9x9=81 CAN be a praxeological statement, it doesn't have to be. so not even all basic mathematics is contained in praxeology.>
but what is the point in knowing there are a certain number of donuts, if not to guide one's actions? any practical use of 9x9=81 will eventually have to be phrased or thought such a way that the action element is explicit in the statement/thought. i can say "there are 81 donuts" with no action element, but once anyone goes to use it for any purpose they will effectively be thinking, if not speaking, of it in a praxeological way, such as "i know i need 80 donuts. i also know this donut shop sells them in boxes of nine. since 9x9=81, i know if i obtain nine boxes of donuts, i will have enough." some might say "well 9x9=81, so i will have enough" but they are at least thinking the underlined praxeological if-action-is-taken-then-a-true-by-definition-result-will-happen statement.
Clayton:I think the issue merits further thought, however. In this lecture, Steven Pinker argues that human language actually contains a latent theory of physics within it and he delves into grammatical constructs to illustrate this. He points out that we say that an ant crawls along the edge of a plate but across the surface of the plate. You do not say "the ant walked along the plate" if you're trying to communicate that the ant walked across the surface of the plate. The mind is dividing the plate into idealized 1-dimensional and 2-dimensional surfaces and rejects the usage of the word along when referring to the act of crossing a 2-dimensional surface. You do not say "he walked along the floor" you say "he walked across the floor."
that's fascinating. language seems to be embedded with a basic set of epistemological and physical assumptions, but they are shared as you say. it's kind of like sharing a car, sometimes other people don't go where you want to go, or aren't as careful drivers as you'd wish, but you can't get out of the car (speak your own language that matches your personal assumptions about how the world works) or else you'll never get to where you want to go (communicate with others).
i heard you say you work in software. do you think there is any way something like this point can, actually or metaphorically, carry over to programming languages? something like, the basic choice of programming language creates a "universe" in which to work to build something...or maybe that certain types of systems are easy to build in one language versus another, perhaps partly because the language was designed with some latent theories of how things work (or should work) in mind?
baxter:>if we are not talking about bending in any direction, in what sense are we saying that a long, thin pole "bends"? A pole in a gravitational field bends in the same way that a geodesic on the Earth bends. A flat-earther walking along the geodesic might not perceive the bend, but a large-scale geometrical study would reveal it.
A pole in a gravitational field bends in the same way that a geodesic on the Earth bends. A flat-earther walking along the geodesic might not perceive the bend, but a large-scale geometrical study would reveal it.
geodesic meaning the shortest path between two points? if so, walking along the earth from Wales to Stockholm does not really trace out a geodesic, as the shortest path would be to burrow through the earth.
or geodesic meaning a curved path along the surface of the earth? then we can say which direction it bends: downward from the POV of the person walking.
baxter:Indeed, the possibility and meaning is imbued in the definitions. Just like defining the successor operation and saying 0'=1 assumes that its even possible or meaningful to have numbers other than zero. Your insistence on "existence proofs" makes you sound like a constructivist-type mathematician http://en.wikipedia.org/wiki/Constructivism_(mathematics). I'm not sure how far that mentality can be carried, since presumably they depend on axioms as well.
yeah i have constructivist sympathies. for natural numbers, i'd define them as a symbol representing a count of 20, or 7 iterations of a process, or 5 pillows. existence proofs:
that was a long post, I. Ryan, but i thought this gem deserved its own hearing:
that bit was, like, double-ultra insightful!
as to the part about 3D is only sets of 2D, i think there is more to it. if we never move, and nothing around us ever moves or changes, we only see 2D faces of everything, but we still perceive more than that. more than we are seeing. i would almost say we are feeling something while we are seeing:
Zangelbert Bindledack: as to the part about 3D is only sets of 2D, i think there is more to it. if we never move, and nothing around us ever moves or changes, we only see 2D faces of everything, but we still perceive more than that. more than we are seeing. i would almost say we are feeling something while we are seeing:
I think that it was David Hume who said that, even if we were to suspend ourselves in a cage 10 stories up with absolutely no danger of falling, we would probably still feel the intense emotions associated with being in danger of falling 10 stories down. But we must realize that, if we don't have the idea of 3D objects, that 2D image isn't anything but one that our body is hard-wired to say is really serious. (I'm not going to say that it's something that our body is hard-wired to say that we really don't want it or something, because some people enjoy putting themselves in those situations.) I can "feel" something really strong even just looking at that picture, but the only way that I know that it has anything to do with 3D objects is because I have seen that running through certain sets of 2D objects before or after getting that feeling is "causally connected" with it. It's like trailing. Sure, we're hard-wired to see some 2D frames as having the "trailing" thing coming after it, as if it's moving, but I only know that it's "as if it's moving", because I see that the 2D frames which generally come before or after those trailing 2D frames are usually... well, you know.
So it's simply that we know through experience that certain 2D frames, or even certain bodily sensations, are associated with sets of 2D frames which form our ideas of 3D objects, that makes it so seeing just one 2D frame, or feeling like one emotion while looking at just one 2D frame, gives us a glimpse into the 3D objects. But, then again, it's nothing different than the fact that seeing a snapshot of your hand with the "trailing" thing going to the left makes it obvious that your hand was moving to the left. Was it that the other 2D frames were actually "contained" in that 2D frame, so you could "see motion in just one frame"? No, it's just that you know through experience that seeing that snapshot in your mind generally follows seeing the other snapshot in your mind, so your mind simply transitions from the snapshot of your hand making to trail, to its "usual attendant". It's just experience that lets us know that certain 2D frames are associated with certain sets of 2D frames making up 3D objects! Without the experience, they wouldn't be anything but 2D frames!
(Start from that still image with left trailing, and then try to think about it moving from the left to the right without your mind transitioning from one 2D frame of your hand farther to the left, to another one of your hand farther to the right. It's not possible! And start from that 2D image that you posted, and then try to think about the fact that you're really far up, or that the cars down there are moving in 3D space, without letting your mind transition from one 2D frame - where you're "higher", to other ones - where you're "lower". It's just not possible! When I think of the fact that I'm really high up in that picture, I imagine sky-diving headfirst off of the top, and then right at the end dipping upwards as to follow the path of the cars. That's a sequence of 2D images!)
But how do I deal with the fact that touch and proprioception show us 3D objects directly? Well, I don't, because I don't think that they actually show us 3D objects directly. I don't want to make this too long, because I haven't even managed to explain how we form our idea of 3D objects yet, but I want to point in the right direction. Basically it's this. Grab your mouse, and feel your hand wrapped around its 3 dimensions. What's going on there? What I feel is a mild "touch" sensation with a smooth texture (I'm not sure whether texture is basic or not yet) distributed around the 3D object. But what's it mean to be distributed around the 3D object? Basically I don't think that it would mean anything until we know what 3D objects are! I can't contemplate my hand being wrapped around the mouse without seeing my mind transition between the 2D frames making up my idea of the mouse. It's just like sound. Sure, you could say that we directly see 3D objects through sound, because we hear that a sound comes from farther or closer. But really what is the mechanism that makes it seem like sounds are coming from a certain location? Certainly the sound itself doesn't really have a location, considering that you might hear a phone ringing jumping around the room, first located on your cellphone, then on somebody else's, then finally settling on a scene on your television. What happens is that we already have our idea of 3D space (which is nothing but sets of 2D images categorized in terms of our desire toward them!), and our mind just transitions so easily from the sound (which has no place), to the 3D object that we feel caused it, that we're apt to confound them, until we feel as if the sound has a place. Same with touch, and same with proprioception. But what happens if you're born blind? How do you develop your ideas of 3D objects, so you touch and proprioception could be distributed along 3D space? I have no idea. Maybe you can tell me. But, either way, I was just talking about how touch and proprioception (and sound!) pretend to show us 3D objects directly, not how they would show it to us through a sequence of frames. I'm almost certain that they can't show it to us directly, but I'm sure that they could show it to us indirectly, through a sequence of frames.
how we perceive 3D objects is out of my realm of investigation so far.
...so what follows on this topic is entirely SPECULATION by me...
first i will say i think the problem has to be bite-sized. start with something VERY simple. otherwise there is little chance we can penetrate this. it's just too big and complicated an issue.
atheists argue endlessly with theists about whether god exists. the issue isn't "god", the issue is "exists". define exists and you'll know the answer to this age-old question. the reason it is age-old is no one ever does this, because they are stuck on the "god" part.
in the same way, we are talking about how we comprehend 3D objects. what are they, and so on. so far discussion has focused on the 3D, but the real action i think is in what an object is, or what we are trying to get at when you call something an object. clarify that and the rest might fall into place. otherwise we could go back and forth for pages talking about whether god exists without ever defining exists, and that would suck.
so, what is an object to you in this case? i don't even have an answer yet. i haven't thought about this, unlike the rest of what i wrote so far in this thread.
by the way, i realize the falling sensation is instinctive and all. i chose that image for maximum impact, but i think it can be shown more mundanely.
something that is farther than, say, arms' length from you might have a different feel from something closer...and of course it depends on what that thing means to you, and to your ability to grasp it, etc. some of it is probably instinctual, so it could be quite a messy problem to fully get one's head around.
Zangelbert Bindledack: atheists argue endlessly with theists about whether god exists. the issue isn't "god", the issue is "exists". define exists and you'll know the answer to this age-old question. the reason it is age-old is no one ever does this, because they are stuck on the "god" part. in the same way, we are talking about how we comprehend 3D objects. what are they, and so on. so far discussion has focused on the 3D, but the real action i think is in what an object is, or what we are trying to get at when you call something an object. clarify that and the rest might fall into place. otherwise we could go back and forth for pages talking about whether god exists without ever defining exists, and that would suck.
Absolutely excellent point.
Zangelbert Bindledack:
That's a sequence of 2D frames, but I see something similar from one frame to the next. It isn't that, with every new frame, I see something totally new, but that, with every new one, I see that it changed slightly. But how does that happen? Well, let's start out with something a bit easier. Imagine the same picture, except all of the balls are gone except for the red one. There's the blue background, and the red ball is moving about it totally unobstructed. So how do we see the red ball as something separate from the background? No idea yet. But I think that we will need to answer that question before we can get anywhere with this.
So how do we see the red ball as something separate from the background? No idea yet.
I think we have some idea. First, we have the ability to sense a distinction between red and blue (necessary but not a sufficient condition for distinguishing the "red ball" from the "blue background.") The next problem is much more difficult, philosophers have been struggling with it since the time of Plato - how does the mind validly correlate a bunch of particulars into a single universal? If we look at the red-ball-only movie frame by frame, these are the "physical facts". Yet when these frames are played in order before your eyes, you perceive something more than a series of unrelated movie frames, like a family genealogy slideshow.
I think, today, we can understand this process a little better through Solomonoff's theory of induction. Basically, I think the mind is constantly formulating hypotheses about the "raw data" which is coming into the brain, searching for patterns that it is able to recognize (it's not able to recognize all patterns, only some patterns). The constructed hypotheses are what create the sensation of wholeness to a series of otherwise unrelated physical sensations.
Some patterns are learned, for example, the multiplication table. There is clearly no evolutionary or biological basis for recognizing the patterns in a multiplication table. But patterns that are reflections of the physical world, like the billiard ball frames in your post, probably have dedicated "pattern recognition" circuitry in the brain.
do you think there is any way something like this point can, actually or metaphorically, carry over to programming languages? something like, the basic choice of programming language creates a "universe" in which to work to build something...or maybe that certain types of systems are easy to build in one language versus another, perhaps partly because the language was designed with some latent theories of how things work (or should work) in mind?
That's a good thought. I've actually not made the connection between latent theories of the world (social/physical etc.) that Pinker discusses and programming languages. Mathematically speaking, all computer languages of a certain 'power' are all equivalent since each can, in theory, simulate all the others. Nevertheless, it is certainly easier to do some tasks in one language (matrix transpose in Matlab) and other tasks in another language (text pattern matching in Perl), etc. I'm designing my own computer language as a hobby... you've just sent me back to the drawing boards, damn you!
P.S. I'm in computer engineering (not quite the same thing as software, but close).
Clayton: I think, today, we can understand this process a little better through Solomonoff's theory of induction. Basically, I think the mind is constantly formulating hypotheses about the "raw data" which is coming into the brain, searching for patterns that it is able to recognize (it's not able to recognize all patterns, only some patterns). The constructed hypotheses are what create the sensation of wholeness to a series of otherwise unrelated physical sensations.
(Warning: Don't try to read this unless you're really patient or something.)
Imagine that the red ball is right in the middle of the picture, and then try to predict where it will go next.
It will move one unit of distance - whatever that means - in one of the whatever amount of directions there are. We can't narrow the options down to anything but that it will only move one unit. It could move up, down, left, right, up/right, up/left, or whatever. We have no idea what direction in those 360 degrees it will "choose". But what if we then see it move one unit to the right? Now we know that it can't do anything but keep moving one "unit" to the right until it hits the right side of the picture and then bounces back toward us.
But let's go back to the first frame, where we don't know anything but that it's sitting right in the middle of the picture. We just talked about what will happen after that frame, but what about what will happen before it? Do we know anything about what the frames were that would have came before it, if we had experienced them? Do we know what we would have experienced if we were to have been there to see what happened before that first frame? Well, we can narrow it down a bit. We can say that the frame directly before the frame that we see right now would have shown us the red ball one unit of distance away from where it is right now, or simply where it is right now. We're able to know that the frame right before the frame that we're experiencing right now was either the ball one unit away from where it is right now in any possible direction, or just the ball in the same place that it is right now. In case this is getting convoluted, let me state it in more natural terms. Either the ball came from somewhere adjacent to where it is right now, or it was just always sitting there. But now let's imagine that we experience another frame, and it's the ball one unit adjacent to where it is right now. That means that we can throw out the possibility that the ball was just sitting there, and we get back to where we were in the first paragraph.
But let's talk about those two frames again. Let's say that the two frames that we have experienced are the ball right in the middle of the picture, and the ball one unit to the right of there. We know that its only course of action could be to continue to the right. Its only option is to keep going toward the right-most edge of the picture, right? No, on second thought, that's wrong. I came to that conclusion in the first paragraph, but it's not right. Let me explain why it's wrong. Well, first, let me say what the real prediction would have to be. We couldn't say that the ball is definitely going to continue to the right, but only that the ball either went from the left to the right and will continue to the right, or that it went from the right to the left, and will continue to the left. But what the hell? Doesn't that mean that it would have to go from the middle, to the right, and then jump to the left? No, I smuggled in some baggage when I said that "the second" thing that we would experience. Ah, but how do we know that it was the second thing? What do you have in your memory? In your memory, you don't have anything but a bunch of frames. Your memory is what puts them in a sequence. Haven't you ever had the experience of not remembering what just happened, simply because it was so similar to every other time, that you couldn't put it in a sequence? Okay, that was terribly explained, so let's start over. Let's look at an example. Imagine opening your refrigerator, looking through the entire thing, and then pulling some lettuce out. Now imagine asking yourself whether you had any oranges left. Well, maybe you don't know. But how do you not know? You just looked 3 seconds ago! Well, the problem is that you open your refrigerator so often that there really wasn't anything "different" about that frame then anything else. Maybe if something was weird, like you saw that some juice spilled, you would be able to remember whether the oranges were there simply by locating the frame where the juice was spilled, and then asking whether the oranges were there. But, if there's nothing to set it apart - nothing to let you put it in a sequence, you'll be at a loss.
Okay, this is getting really convoluted. Let me start that paragraph over, but with the new insight that we have to put the frames in sequence in our memory based on experience, and not on any original quality of the frames. The frames don't order themselves. You have to order them for yourself, based on the regularity. Just like the direction of time comes from the regularity in the sequence of things growing and deteriorating, the order itself comes from the regularity in general. Okay, let's get back to it. Let's say that the two frames that we have experienced are the ball right in the middle of the picture, and the ball one unit to the right of there. We don't know what the order is; all we know is that we remember two frames, where one is the ball being right in the middle, and the second is the ball being one unit to the right of the middle. Let's ask two questions: What happened before that, and what will happen? What happened before it: Either (1) a frame where it's in the same place - it never moved, (2) a frame where it's one unit to the left - it moved from the right to the left, or (3) a frame where it's one unit to the right - it moved from the left to the right. And what will happen after: Either (1) a frame where it's one unit to the left - it will move to the left, or (2) a frame where it's one unit to the right - it will move to the right.
But what happens when we get the third frame? Well, everything falls into place. If it's one more to the left, then we know that the ball came from the right and will move to the left; and, if it's one more to the right, then we know that the ball came from the left and will move to the right. That's it. But where the hell did I get this idea that things can move only one "unit" at a time, and that they can't change direction unless they hit something else? How did I smuggle that in? Well, I have no idea, but this whole analysis has rested on that "hypothesis": That things move in straight lines and at uniform speeds unless something interferes. With that hypothesis, 3 frames it all that you need to know exactly what happened and will happen, in our simplified model. But, again, where the hell did I get it? Well, I guess that I would have had to experience a certain amount of situations where only that happened. Or something.
"but i actually will give a counterexample that seems to go against this in order to make the point:
<9x9=81 could be used to learn something completely unrelated to human action: i know there are 9 boxes of 9 donuts each in my kitchen, so i can know there are at least 81 donuts there. "there are 9 donuts" is not an action, so although 9x9=81 CAN be a praxeological statement, it doesn't have to be. so not even all basic mathematics is contained in praxeology.>"
ZB:
Going back to your original statement that "at least most of mathematics is best demarcated as a sub-branch of praxeology," I would argue that this notion is more comprehensive than you may realize, for the following reasons:
It is not necessary that "there are 9 donuts" be an action for "there are 9 donuts" to be subsumed by praxeology.
We can easily conceive of an action as being comprised of both a means and an end. Means and ends are components or categories of action.
Then, to argue that "there are 9 donuts" is not part of praxeology, it will be necessary to demonstrate or prove that "there are 9 donuts" is neither a means nor an end of action. If "there are 9 donuts" is either a means or an end of action, then of course "there are 9 donuts" is, by definition, subsumed by praxeology.
Your proposition that not all of basic mathematics is contained in praxeology is based on your premise that "there are 9 donuts" is not an action. But if "there are 9 donuts" is either a means or an end of action, then "there are 9 donuts" is contained in praxeology.
Also, for any object X we may assume or suppose, it may not be necessary to answer the ontological question as to what X "is".
Instead, it may only be required that we consider X "as" a certain thing, or consider X from a certain point of view.
I.e., we may consider X as a means of action, or as an end of action. We may consider X as something an actor is trying to obtain or attain, or consider X as something the actor is utilizing as means.
Then, to argue that X is not part of praxeology, it will be necessary to prove that it is impossible to consider X as a means or end of human purposive activity.
One of the most important passages in Mises touches on this:
"It is of primary importance to realize that parts of the external world become means only through the operation of the human mind and its offshoot, human action. [External objects] are as such only phenomena [of the physical universe] and the subject matter of the natural sciences. It is human meaning and action which transform them into means. Praxeology does not deal with the external world. but with man's conduct with regard to it. Praxeological reality is not the physical universe, but man's conscious reaction to the given state of this universe. Economics is not about things and tangible material objects; it is about men, their meanings and actions. Goods, commodities, and wealth and all the other notions of conduct are not elements of nature; they are elements of human meaning and conduct. He who wants to deal with them must not look at the external world; he must search for them in the meaning of acting men." (Human Action)(emphasis and brackets added)
When Mises writes that human meaning and action transform the external objects of the physical universe into means, he doesn't mean this in the ontological sense (that in utilizing objects as means, we transform these objects physically). What he means is that for any given object X, praxeology as the theory of action, looks upon object X as a means or as an end of action.
Praxeology need not be founded upon nor preceded by a process of ontological categorization:
Object X "is" the subject of praxeology. Object Y "is not" the subject of praxeology.
Object X "is" a conscious being. Object Y "is not" a conscious being.
Praxeology can be approached as a purely formal endeavor:
If object X is a means, then qrs. If object X is an end, then tuv.
If object X is an acting being, then xyz laws apply. etc...
Clayton:The next problem is much more difficult, philosophers have been struggling with it since the time of Plato - how does the mind validly correlate a bunch of particulars into a single universal?
it doesn't. particulars are lumped together and used that way1, with a few hacks and tricks to approximate generality.
we can then take that bundle of particulars and find certain characteristics common to all (or most) of them, then use those characteristics as guidelines for determining what new particulars should be placed in the category.
in the case of the moving ball, though, i'd guess we have some built-in machinery for recognizing motion and object permanence, as you alluded to. though since part of it is hard-wired in and probably part is learned in childhood, a complete explanation would get messy.
1with this showing up all over the place in typical human patterns of error
Clayton:That's a good thought. I've actually not made the connection between latent theories of the world (social/physical etc.) that Pinker discusses and programming languages. Mathematically speaking, all computer languages of a certain 'power' are all equivalent since each can, in theory, simulate all the others. Nevertheless, it is certainly easier to do some tasks in one language (matrix transpose in Matlab) and other tasks in another language (text pattern matching in Perl), etc. I'm designing my own computer language as a hobby... you've just sent me back to the drawing boards, damn you!
if you could elaborate on that somehow when you've considered it more, i would look forward to it. could methodological individualist assumptions versus collectivist ones, or some similar distinction, result in a different programming language (like for an economics simulator1)? and in what sense is a programming language like a spoken language?
1i really have no clue about this, so make the necessary adjustments to something that would make sense
I. Ryan:In your memory, you don't have anything but a bunch of frames. Your memory is what puts them in a sequence.
you seem to be suggesting that your memory puts them in sequence for logical or practical reasons, but it seems to me that our memory faculties would just automatically have some method of keeping track of which thing came first and which came next.
this is a bit thorny because we have to keep track of our assumptions.
and what is an ordering? so as not to smuggle in the concept of time through the back door, i would say it is an certain type of utility prioritization, at least. that's something as a start; a promise someone made to us "long ago" is regarded as less relevant to our current actions than a promise "a moment ago". so in one sense it is a prioritization scheme, where we call "more important past happenings" the recent past, and "less important past happenings" the distant past. but that is just one aspect of it.1 what are the rest?
this is why time has been a hard concept to iron out: if we try to do it from the individual perspective, which is ultimately the only valid epistemology, there are a lot of different things going on and going into the concept we summarize in a single word: time.
1and i realize this could be reversed in some cases, but there is a general tendency like that...time heals all wounds, and such.
>if we are looking from the individual perspective, we can know nothing of any objective notion of time in the natural universe.
Unless you grow up in a closet, you have to deal with physical processes that are ordered in time. You have to deal with other people's plans and actions when formulating your own plans and actions. I find it hard to imagine a non-invalid person who is unaware of an objective ordering of physical events that corresponds with their memory ordering and with everyone else's memory ordering.
Even growing up in a closet, you still have a chance of learning the objective ordering of events. For, your own body is a datum external to the human mind and it obeys physical processes.
BTW, I'm talking about the time-like intervals of everyday experience. Space-like intervals have no objective ordering (per relativity theory). I cannot cause something to happen or prevent something from happening 1 light year away in less than a year's time, no matter how clever or powerful I am. Such an event occuring is objectively neither in my past nor my future; it is outside of both my memory and my influence. Space-like intervals do not involve cause-and-effect; such events are only data to be learned after-the-fact.
Adam Knott:But if "there are 9 donuts" is either a means or an end of action, then "there are 9 donuts" is contained in praxeology.
'guess if you take the idea of a means to its logical end, it really could be the entire state of affairs (Mises) someone is faced with. an axe is my means to chop wood...but really so is gravity, friction, momentum, my arm, the fact that my arm is attached to my body, the fact that i am not in handcuffs, the fact that there is light for me to see by....these are all means if any of them is! why the axe any more of a means than the light?
still, all of those are necessary for me to chop the wood. there being 9 donuts in my car is not necessary for my chopping of the wood. i don't utilize that aspect of the state of affairs as a means...but then, is it really an aspect of my state of affairs? or, in such cases, is it just an irrelevant nothing?
ok wait, i think i got something: "there are 9 donuts" has meaning to me in the present moment only as far as it is relevant to whatever action i am taking now. specifically, if i just try to collect random facts that may at some point be useful to me, and this seems a habit of most humans, "there are 9 donuts" is going to be relevant to me for chopping the wood...orrrr maybe not?
nah, this is it: eureka! (maybe). it's really that there's this state of affairs, whole and undifferentiated. it is state of affairs A, and it includes an unchopped block of wood, a whole slew of other things...and 9 donuts in my car. i want to change this state of affairs to a state of affairs B that includes a chopped block of wood, that same whole slew of other things...and that same 9 donuts in my car. i don't care about the 9 dounts really, but if we look at the entire state of affairs as whole and undifferentiated, then we cannot say anything other than that i am "utilizing" or "working" or "acting on" state of affairs A to change it to state of affairs B. then all of A is the means and all of B is the end. that is with no subdividing of the two states of affairs down into specific parts like the axe or the light or my arm or the donuts. then once i reach B, it becomes my new means, and i select a new action to reach state of affairs C, my new end. and so on. or something like that?
it is as if the current instantaneous frame of your experience is your means, the entire thing, not any one part of it, and the next (or next next next...) frame of your experience that you anticipate or hope your present action to accomplish is your end, and each end achieved becomes a new means because it is now your present frame of experience that you will act on to try to get something even better, or at least less worse.
in some ways it's a very unenlightening way to think of means and ends, but it could perhaps have its uses?
Zanglebert Bindledack: if we are looking from the individual perspective, we can know nothing of any objective notion of time in the natural universe.
if we are looking from the individual perspective, we can know nothing of any objective notion of time in the natural universe.
What's an "objective notion of time in the natural universe"?
Zanglebert Bindledack: so as not to smuggle in the concept of time through the back door, i would say it is an certain type of utility prioritization, at least. that's something as a start; a promise someone made to us "long ago" is regarded as less relevant to our current actions than a promise "a moment ago". so in one sense it is a prioritization scheme, where we call "more important past happenings" the recent past, and "less important past happenings" the distant past
so as not to smuggle in the concept of time through the back door, i would say it is an certain type of utility prioritization, at least. that's something as a start; a promise someone made to us "long ago" is regarded as less relevant to our current actions than a promise "a moment ago". so in one sense it is a prioritization scheme, where we call "more important past happenings" the recent past, and "less important past happenings" the distant past
Interesting suggestion.
Zanglebert Bindledack: you seem to be suggesting that your memory puts them in sequence for logical or practical reasons, but it seems to me that our memory faculties would just automatically have some method of keeping track of which thing came first and which came next.
Have you ever been somewhere so many times that your memories of being there run together such that you can't pick out a frame from your memory and have any idea when you experienced it?
If you only went somewhere once in your life, and you remember slipping and falling in that setting, you can go to the next step and ask when you went there. But, if you went somewhere 10,000 times, and you remember slipping and falling in that setting, what the hell is there for your memory to latch onto? Absolutely nothing. But what if you see something weird, such as somebody dressed as an 80's tennis player in the sequence of frames? Well, then your memory might have something to latch onto.
If the ordering were native to the frame, that wouldn't be a problem. But the ordering isn't native to the frame. We order our memories based on their regularity. When do people dress like 80's tennis players? Halloween! We learn that through experience, so we know that seeing that person in the sequence of frames where we slipped and fell makes it likely that it happened on Halloween. And then maybe you remember leaving somebody's house (where a pumpkin was on the front step) and then getting to the place where you slipped. Oh, that means that it was on the Halloween that I went to that person's house... and so on, until you recreate a simplified sequence up to the current time.
But, wait, I just smuggled in an order right there. I remember leaving their house and then getting to the place where I slipped? Isn't that an order? How did I know that the frame where I was leaving their front step was before the frame where I slipped? Either by the same method, or maybe there's something automatic at the lower levels. Not sure yet.
in what sense is a programming language like a spoken language?
Well, the word "language" is somewhat misleading in the phrase "programming language." Obviously, computers don't "speak" or even communicate with one another in the way that humans or even other animals do.
A programming language is only a language in the technical, computer-scientific sense of being (in the most general sense) a subset of all strings drawn from the "Kleene-star" (length-wise lexicographic combinations) of a finite alphabet. Languages are frequently defined in terms of acceptance by a machine, that is, "L is the language of all strings accepted by machine M." A machine need not be physically instantiable (i.e. it can have some infinite resource, such as an infinite tape, etc.) but every machine I'm aware of is physically comprehensible, that is, you can imagine the operation of the machine as a physical device, locally.
There are a wide variety of machine types (and a wide variety of language types) but the most general and powerful machine is the Universal Turing Machine (UTM). Machines which are superficially different in their construction but which can be proven to be formally equivalent (and, thus, to accept the same language) are said to be equivalent and have the same "power". Any UTM can simulate the operation of any other UTM as well as the operation of any formal machine (a UTM accepts the "Universal Language" which contains every other formal language as a subset).
I said all of that to be able to say that a programming language can actually be thought of as a machine specification. At least, it specifies what kind of machine is required. In the case of the Universal language, the machine specification is "any UTM or equivalent".
But that's all really abstract theory. Programming languages are rooted more in practice than in theory (though each individual programming language may borrow heavily from one or more aspects of computing theory). A programming language is almost invariably a transform description, that is, it describes how to transform a set of high-level instructions into a functionally equivalent lower-level set of instructions. For example, if you write a program in C#, it will first compile down to Microsoft's Intermediate Language (IL), which can then be compiled further down to machine instructions (which are a language which the physical CPU can read and execute). Perl6, similarly, compiles down to Parrot instructions (PASM), which can be run on a "native" (machine language) interpreter that dynamically converts the PASM code to native machine code. Java uses a method call JIT-compilation (Just-In-Time compilation) which is a nifty little trick for squeezing more performance out of interpreted code*.
That's all background. To get back to your question, programming languages are like spoken language insofar as they are a machine specification. Think of engineering an automobile engine. Somehow, you have to describe all those curved surfaces, the join points, dimensions, functional behavior, and so on. A programming language implies an abstract machine which has a sort of "mental shape"** to it, and it is this "mental shape" that constricts or liberates the programmer by making some things easy and other things hard. Not everything can be easy... unless you've invented AI (lol). However, a program language designer can invest sweat equity into his language and save the users of his language the effort of repeating that sweat labor. This is a lot harder than it sounds because you cannot foresee how your language will be used in the wild. You might solve a particular problem in your programming language and it turns out that people want to solve a problem just slightly different than the one you solved for them. In which case, they will simply neglect all that labor you poured into your language (and, if your language isn't buying them anything else, they might just abandon it for something more useful). Or, you might solve the problem that everybody needs solved, but then you package it in an alien way that programmers find counter-intuitive, backward or otherwise un-aesthetic and they will simply neglect to use it, even though it's right there for them to use. This is not so much different than the problems facing UI (user-interface) designers for operating systems (or microwaves or DVD players, for that matter).
*The old-fashioned bright-line distinction between compiled code and interpreted code is blurred in modern languages
**The "mental shape" is the implied, abstract machine residing behind the semantics of the language. In C, for example there are these things called "pointers" but pointers assume a certain memory model that is not applicable in other kinds of languages. This gives C a unique "mental shape" from other languages which do not have pointers.
Zanglebert Bindledack: nah, this is it: eureka! (maybe). it's really that there's this state of affairs, whole and undifferentiated. it is state of affairs A, and it includes an unchopped block of wood, a whole slew of other things...and 9 donuts in my car. i want to change this state of affairs to a state of affairs B that includes a chopped block of wood, that same whole slew of other things...and that same 9 donuts in my car. i don't care about the 9 dounts really, but if we look at the entire state of affairs as whole and undifferentiated, then we cannot say anything other than that i am "utilizing" or "working" or "acting on" state of affairs A to change it to state of affairs B. then all of A is the means and all of B is the end. that is with no subdividing of the two states of affairs down into specific parts like the axe or the light or my arm or the donuts. then once i reach B, it becomes my new means, and i select a new action to reach state of affairs C, my new end. and so on. or something like that? it is as if the current instantaneous frame of your experience is your means, the entire thing, not any one part of it, and the next (or next next next...) frame of your experience that you anticipate or hope your present action to accomplish is your end, and each end achieved becomes a new means because it is now your present frame of experience that you will act on to try to get something even better, or at least less worse.
When you say "everything", do you mean just everything in the outside world, or do you mean everything not only in the outside world, but also in your head?
quickly: i didn't use the word "everything" in that quote so i'm not certain i know what you mean, but i guess i would mean neither "everything in the world" nor "everything inside and outside my head"; i would just mean everything in my head (because i am assuming the methodological individualism point of view).
but also, that is just me trying to see if i correctly understood AK or not, so it is all provisional.
no time for a long reply now, but while i'm at it: my "objective notion of time in the natural universe" is, strictly speaking, nonsense from the individual point of view. i mentioned it only because it is of course the common view and i have to deal with that to clarify the discussion.
Sorry, I wasn't clear enough.
Zanglebert Bindledack: nah, this is it: eureka! (maybe). it's really that there's this state of affairs, whole and undifferentiated. it is state of affairs A, and it includes an unchopped block of wood, a whole slew of other things...and 9 donuts in my car. i want to change this state of affairs to a state of affairs B that includes a chopped block of wood, that same whole slew of other things...and that same 9 donuts in my car. i don't care about the 9 dounts really, but if we look at the entire state of affairs as whole and undifferentiated, then we cannot say anything other than that i am "utilizing" or "working" or "acting on" state of affairs A to change it to state of affairs B. then all of A is the means and all of B is the end. that is with no subdividing of the two states of affairs down into specific parts like the axe or the light or my arm or the donuts. then once i reach B, it becomes my new means, and i select a new action to reach state of affairs C, my new end. and so on. or something like that?
Does that "state of affairs whole and undifferentiated" include just what's in the outside world (the donuts, the wood, and so on), or does it also include what's in your head (imagining yourself chopping the wood, some bodily sensations, and so on)?
Zanglebert Bindledack: i would just mean everything in my head (because i am assuming the methodological individualism point of view).
i would just mean everything in my head (because i am assuming the methodological individualism point of view).
Rough:
You can build your idea of the outside world (the idea that things exist whether you're there or not, that you can be wrong about what you think is there, and so on) out of the methodological individualism point of view. (When I say "in your head", I just mean everything that you don't consider a part of the outside world. But of course the outside world for you (I'm guessing...) is just a model for predicting what 2D image, bodily sensation, or whatever will come next.[EDIT: and whether you'll like it!])
yep both those. although i am noncognitive as to "the world", that is i guess not the point of your question. yah, everything i am experiencing is my state of affairs, so that would include every impression that i am enjoying or un-enjoying, whether i classify it as coming from within or without.
(sorry others, i will respond in full later)
Zanglebert Bindledack: yeah both those. although i am noncognitive as to "the world", that is i guess not the point of your question. yeah, everything i am experiencing is my state of affairs, so that would include every impression that i am enjoying or un-enjoying, whether i classify it as coming from within or without.
yeah both those. although i am noncognitive as to "the world", that is i guess not the point of your question. yeah, everything i am experiencing is my state of affairs, so that would include every impression that i am enjoying or un-enjoying, whether i classify it as coming from within or without.
I'm not ready to substantiate this, but I suspect that the ideas of "choice", "means and ends", and so on don't make any sense except when you're referring to your interface with the outside world.* (Which would mean that your "state of affairs" - your means - would be just what's in the outside world. If you look at everything - including what's in your head - there doesn't seem to be any "choice" left. I just find by the principle of association that my mind transitions from X to Y etc and that's it. Perfect determinism.)
*Unfortunately you could probably interpret that like 50 ways that I'm not talking about.
Don't worry about responding to this now if you don't have time right now...
Back to infinity for a moment.
The natural numbers can be defined by positing a "first number", 1, and a successor function:
s(x) = x+1
s(1) = 2
s(2) = 3
etc.
To me, infinity is the inevitable result of the following question - is there a number large enough to break the successor function, i.e. is there a largest number, or a number which does not have a successor? I think the answer to this is clearly no.
The next question which naturally follows is, how many numbers are there? The answer, in terms of numbers we have already defined, is "no number". There is no number that says how many numbers there are. But when we list the numbers out: 1, 2, 3, 4, 5... they look perfectly countable. So why can't we count the whole lot of them, just as if they were a really big barrel of apples? I see no reason why we can't define a symbol, say Aleph_0 and say, "The number of natural numbers is Aleph_0" especially since there's lots of interesting things we can then do with this symbol.
Of course, such a magnitude cannot be physically instantiated... yet neither can a geometric plane and nobody has a problem with that. A geometric plane exists only in our imaginations... it's a shared illusion or shared mental abstraction. Same goes for infinity.
Clayton: The next question which naturally follows is, how many numbers are there?
The next question which naturally follows is, how many numbers are there?
I think that is the clearer question.
To me, infinity is the inevitable result of the following question - how many numbers are there?
The answer perfectly consistent with Zang's posts is this - as many as you want there to be.
as many as you want there to be
That's easily disproven. I want there to be 3 natural numbers.
{1, 2, 3, 4}
QED
Clayton: That's easily disproven.
That's easily disproven.
Okay, let's see it.
Clayton: I want there to be 3 natural numbers.
I want there to be 3 natural numbers.
Solid so far.
(Apparent you find it useful to have 3.)
Clayton: {1, 2, 3, 4}
Still solid so far.
(Before you found it useful to have 3, now you find it useful to have 4.)
Clayton: QED
But can't you change what you want?
(Looks like you changed what you wanted from one line to the next.)
I understand the praxeological aspect of this, and assent to its importance, but I feel that it completely misses the point. We (humans) envision the world in certain, definite ways (see this lecture by Steven Pinker here). These abstract, immaterial ways of envisioning the real, physical world are, I believe, the ultimate foundation of most mathematical constructs. What do we mean when we say "a straight line cannot cross itself"? Do we really mean that a rigid piece of physical substance which is straight (whatever that means) cannot cross itself? Or do we mean something more ephemeral and metaphorical? I don't think we're making any kind of statement about the physical world outside of our brain. The formalists try to say we're merely pushing symbols around, concomitant to certain rules (whatever a "rule" is) and "A straight line cannot cross itself" is simply a statement about the properties of any object which can properly be described by the symbol "straight."
However, I reject both of these views. I believe what we're really saying is more like what Pinker is talking about in how we use language to slice up the physical world in very specific ways. What we're really saying is something more like, "Hey, you know how you envision the essence of straightness in your mind? Well, any object you envision in your mind which you would think of as 'straight' cannot cross itself, right?" The rules of the mathematical formalisms that we find "envisionable" and which we, for that reason, find important are not selected arbitrarily, they are selected to conform to these definite ways in which we envision things in our minds.
I think I need to spend some time thinking of specific examples.