Clayton, here is something important to answer:
I think that answering that will get us a lot further in this discussion.
C’mon man, try to make more charitable interpretations.
I’m saying that you have to think in terms of 2D appearances to understand David Hume’s argument.
(If you try to think of it in terms of 3D objects, you won’t get it!)
sure it does. try it! back away from this set of dots …
until it looks something like ________________ except that it will appear dark grey instead of black.
EDIT: i see you meant not just that they would appear as a line, but actually “be” a line. i think we should eventually ask what “actually be” really means, but i will withdraw this line of argument for the moment in the interest of focusing the thread.
a line is a long, thin rectangle. we’re not sophists here, so since we are talking about a visual phenomenon we may as well show the object in question.
EXHIBIT A: a white line on a black background
notice that it is several pixels wide. that is intentional. it is to underscore that the line is not infinitely thin. in fact, “infinitely thin” would be nothing more than blather if it meant anything other than “thin enough that the width is negligible for the present purposes”.
to see “negligible width” in a picture is easy for some purposes, hard for others. when it is hard, such as when the line needs to be super-thin, you can view it in a movie: imagine zooming in on the line and showing it not getting bigger for a long time, then finally at the end of the movie the line has a definite width, as it is being viewed in extreme close-up.
Of course, and that’s what I’m talking about.
We don’t have anything but our various views of the point.
So what is “the point itself”? Well, just the set of our various views of the point!
(And I’m saying that each of our “views” is nothing but a 2D appearance made up of a finite number of indivisible points.)
This I agree with.
This I don’t agree with - or even if I do agree with it, I don’t think you’ve come anywhere near establishing it as a fact.
Clayton -
glad we at least agree here. (this discussion might have little if any bearing on the practical applications of mathematics.)
this one is answered above: https://forum.freecapitalists.org/t/anyone-frequent-any-science-or-physics-forums-groups/16410/82
to re-gloss, infinities are not implicit in the objects or shapes themselves, but rather we can prove that the PROCESS of creating a decimal expansion of pi or the square root of 2 can be continued for as long as we like. and i am fine with using “ad infinitum” as a shorthand for the underlined, as long as we are explicit about it.
indeed, large numbers are famously hard to comprehend. the situation is not as bad as it seems. in fact it is exactly as bad as you thought before you read this, because we are using numbers like 1000 and 1000000 every day without very much problem. we also rely on calculating methods and machines, and do so for series of small reasons each of which can be comprehended, or simply for the reason that they don’t often let us down.
we can’t even subitize beyond 4. that is, the concept of 5 apparently requires memory for some; 4 is a picture, but 5 may need to be a movie. at least, to an untrained child.
there are all sorts of ad hoc tricks we can use to understand what numbers like 1000 mean to us in terms of anticipated experience. they are not perfect, though, and their shortcomings are often revealed to us when our anticipated experience differs from what actually happens. [if we use “infinity” to mean our experience will have an incessant component in it, that would be fine.]
so the short answer is: 1, 2, 3, and 4 are simple pictures that can be conceived in the single frame. some people can conceive of more, or much more in a single frame, using various tricks, and most people seem to be able to conceive of much higher numbers with some semblance of accuracy.
some tricks: counting groups of 4 or 5, using points of reference such as that $20,000 is enough to buy a nice new car, zooming out, using vague notions like “an order of magnitude is a lot!”, etc.
BUT we cannot imagine “infinity” other than imagining something like “way big enough to be big enough for the present purpose” or “don’t expect this process to terminate”.
this approach only works if it is clear that the actual “something” in question is not a nonsense notion. i can talk about “some physical object O” or “some number x” because we know what a number is and what a physical object is. in short, we cannot talk about the properties of something until we at least know what type of notion it is; infinity is not a number, for reasons already explained…so is it a strawberry? is it treason? is a triangle? a unicorn? i see a problem with listing properties of an object whose class is wholly undefined.
Your objections are at least as valid for all numbers other than natural numbers and rational numbers (I think they’re valid for those, as well, since I think I’m a mathematical fictionalist). Does 0 exist? What is it? A strawberry? A unicorn? Does -1 exist? What is it? Is it treason? Does i exist (come on, you have to admit i is really freakin’ weird).
I’ll try to post a section from an essay I’ve written on this very subject.
Clayton -
they are just symbols that mean the same thing, yes, but insofar as they mean the same thing, we haven’t LEARNED anything. the practical import is that going halfway there, then a quarter of the way, then an eight of the way, etc. will get you closer and closer to wherever “there” is. it is only when we create a movie in our heads that we actually learn why the equation is useful.
i can understand what this equation means for my anticipated experience by seeing a movie of a dot that is one foot from a line moving 1/2 foot closer, then 1/4 foot closer, then 1/8 foot closer, until i am satisfied that the longer i imagine the movie the closer the dot will be to the line.
to do more complex convergences, i can use logic to derive the rules of working with limits1, and then just calculate using those rules, with the knowledge2 that the rules faithfully represent movies that could be imagined. but truly, there is the rub. symbols and formal calculations only make sense as long as we believe they represent something that can, in principle, be imagined.3 either something we have imagined before, or something we trust that the experts have imagined for us and enough people have checked it and used it so that we can be reasonably confident it is useful.
1which require no concept of infinity as such
2if i have done my logical derivations correctly
3or if the formalism “just works” and we don’t know why…but this is equivalent to magic, religion, superstition, mysticism then. that doesn’t mean it’s not useful: praying to the skiing spaghetti monster every night could help you achieve your goals faster than otherwise. being right for the wrong reasons is still being right, but it doesn’t bode well for continuing to be right under different circumstances, because you are focused on the wrong underlying mechanism for your success so far.
so it will no longer be triangular, right? and what direction does a sphere near a star “bend”? can you point to that direction?
epsilon-delta is hard to understand? you seem to be using the word infinitessimals to mean “a formal representation of something we have previously understood by the logic of the epsilon-delta process”, so i do not think we are disagreeing.
things that do not actually exist, OR actions that are not actually possible. (or any other word uttered by someone that does not correspond to any consistent thought they have had [or often, corresponds to some other thought entirely that does not work for their theory])
numbers are symbols that represent, at bottom, the act of counting. they aren’t subject to questions of existence or non-existence, any more than any other act is. counting 7 watermelons is a possible action one may take, for instance.
the key difference is that finite numbers can be imagined by visualizing the movie of counting for small numbers, and by various other more-or-less reliable tricks for larger numbers; but “infinite numbers” cannot be imagined, other than as a movie that one doesn’t expect to end. there are no tricks to get us to understand “infinity”, other than the tricks to get us to understand very large numbers, where “very large” means large enough for their diminuitiveness to be negligible in the given context.
again this may not even matter for mathematics in practice, but there is a good reason for talking about it regardless: it certainly may matter for when the formalisms of mathematics are carried over to other fields based on the belief that the mathematicians are super-rigorous. if mathematics is not rigorous in certain aspects because it doesn’t need to be1, that’s fine for mathematics; the problem is when that lack of rigor is transferred elsewhere on the mistaken trust that such careful people as mathematicians have already proved it is valid for us.
SUMMARY: mathematics is only rigorous in the aspects where it needs to be; “infinity” is not one of those aspects (although numbers are treated more carefully by some people and in some textbooks). this may not be a problem for math, but it can be a problem for other fields that appeal to the authority of math, mistakenly believing it to be rigorous in ALL aspects, even those aspects that don’t matter for math done in practice.
1baxter, you asked me to call you when i need to calculate something. i would happily do so. the problem here is not (necessarily) for the field of mathematics, but for those who point to mathematical formalisms that may only be useful in pure math or in certain delimited applications and try to invoke them in other contexts where the formalism would be entirely divorced from reality or utility. that would be an example of a lost purpose.
0 is a bookkeeping symbol, which doesn’t represent anything all by itself. “0 apples” simply means “no apples”. “0 velocity” means “motionless”. “1 - 1 = 0” is a praxeological statement-form that can be applied to create a whole class of valid statements, one of them being “if you pick up one apple, then reverse that action (by putting it back down), you will end up with no net change in your apple holdings”.1
the minus-sign is just a bookkeeping symbol, so -1 means nothing until it has content: putting an apple back down, for instance. processes don’t exist or non-exist, they are instead possible to perform or impossible to perform. at bottom, -1 just means doing the opposite of whatever 1 represents, and of course it can only have that meaning when it is readily apparent what “opposite” means in the context.
if we take the symbol “1” to simply represent a movie of someone counting, then the symbol “-1” would be meaningless because such an act is impossible to reverse or do the opposite of, unless we define it in a special way, like “i owe him 5 apples and have no apples; in other words i have -5 apples” or “i count here 5 apple IOUs, that is, -5 apples”.
finally, even the number 1 has no intrinsic meaning, necessarily, until we say “1 football” or “1 kick”. counting is a natural match because we can understand the idea of counting some fuzzy, unknown objects, but really i should note that counting is always about counting something. (we also call reciting the natural numbers “counting” but that is a different use of the word.)
1is praxeology contained in mathematics or is mathematics contained in praxeology? 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; as i have been trying to show, it all is based on human action and human observation (or the action/observation of any intelligent being).
whoops, i missed this:
incoherence is also the result when someone utters a word that doesn’t match any cognitive action or object in their mind. like “blicksthlezorp”. or even “apple” if the person has no perception or memory of an apple in their mind.
Phew, I see some major differences in views of mathematics. Anyway, Zang, what about if I say that a “blicksthlezorp” is a object that has certain properties? Let’s say that such an object has the same properties as “1”, or in your words, the act of counting one. Can we write “blicksthlezorp” = 1? Can we say this “blicksthlezorp” is the same as 1, or the act of counting one?
Even more in “practical mathematics” than in theoretical we have to say those tho objects are the same since we have no possibility to distuingish them.
Thoughts on this?
1D creatures would only see points
2D creatures would only see lines
3D creatures would only see planes
At every level the creatures would infer a higher dimension. For example a line has no depth, and thus, in a sense, it you could ask how can it see a line of infinitely small depth? Yet, the 2D would creature would be seeing a cross section of a 3D world. They would be inferring depth. But, do we see in planes and infer volume with our memory of the collection of planes?
And I’m saying that each of our “views” is nothing but a 2D appearance made up of a finite number of indivisible points.)
I cannot see this as being true except to the extent that our vision depends on a finite number of biological cells in our eyes. But tricks such as optical illusions demonstrate that one’s subjective visual image is further “post-processed” by the brain/optic nerve. I know of no way for one’s subjective visual image to be shared by another person or converted into pixels.
BTW, I guess I don’t really understand Hume or why it’s important to deal with 2D images. The mind has direct access to 3D geometry through the senses of touch and proprioception.
and what direction does a sphere near a star “bend”? can you point to that direction?
The curvature is a tensor, which can be collapsed into a scalar (i.e. a curvature value): http://en.wikipedia.org/wiki/Ricci_curvature. Physics theory and experiment have no answers as to whether the universe is really 4D or simply a 4D manifold embedded in higher space. In any case, if there were a “curvature direction”, it would point outside of the manifold of our universe, and couldn’t be “pointed to” by an element of the universe.
you seem to be using the word infinitessimals to mean “a formal representation of something we have previously understood by the logic of the epsilon-delta process”
Not at all; infinitesimals have nothing to do with the epsion/delta definition. Infinitesimals are elements that stand on their own both conceptually and formally in the branch known as nonstandard analysis. Furthermore the epsilon/delta definition of limits is NOT a process, but rather contains words like “for each” and “there exists”. Not “as it gets smaller”, etc. It is non-intuitive, awkward, useless in practice, hampers students’ education, and was contrived by people like you who for some reason can’t abide the simple and useful concept of infinitesimals.
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.” Fine, then so is infinity (or, to be more specific so are the well-defined notions of infinity which are part of transfinite mathematics). Note also that you conveniently ignored the imaginary unit, i. Is it a book-keeping symbol? If so, to what does i correspond?
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. I have a great example to illustrate this, it’s called the See-and-Say sequence invented by John Conway. Basically, you start with any number. Then, you repeat the first digit in each run of digits in the number with the length of that run prepended to it. For example:
1, 11 (the first element is “one one”), 21 (there are “two ones” in 11), 1211 (there is “one two and one one” in 21), 111221, 312211, 13112221, 1113213211, …
I’m pretty damn sure there is nothing in the real world which corresponds to the See-and-Say sequence (maybe run-length data compression but that’s artificial, too).
Clayton -
“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)
“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.
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?
I want to give you a better answer, but for now let’s work with this:
We can’t even come up with the idea of “biological cells” or “eyes” without thinking about 3D objects, but we need sets of 2D appearances before we can get to 3D objects, so it’s clear that we need to answer what 2D appearances are before we get to talking about 3D objects.
(How could anybody really disagree with the idea that we see a sequence of 2D appearances? How could anybody really argue that we see depth immediately, like we see the other 2 dimensions, and not through experience?)
Same for this.
Let’s forget about this stuff and get back to what we actually see!
(Until what we see are those things!)
The pixel thing was just a metaphor.
It didn’t really work, so we should probably forget about it.
That pushes the subject radically deeper than my constant one-line responses would suggest.
I will try to get to that later.
I will try to expand on this when I get some time.
mostly yes.1 once you define the word in a way the audience understands, you can use it.
1only thing i would point out is that “1” (interpreted as counting) is a process or act, not an object. [1 watermelon is an object.]
if we are not talking about bending in any direction, in what sense are we saying that a long, thin pole “bends”?
if so, provide a definition. 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 for there to be a number smaller than every positive real number, and (AFAICT) he chugs right along without ever dealing with that issue of possibility. had he later gone on to prove the existence of such numbers, that would be one thing; but in failing to do so, he tacitly attempts to use his definition as an existence proof.
it would be as if i defined “plizzle” as “a real number that is neither positive, negative, nor zero”…and then just kept writing. or if i defined “god” as “god is an omnipotent and omniscient being” and acted as if i had proved God exists…now is this MATH or is it RELIGION?1
1i realize this is not your definition. my exuberance is directed squarely at Ponstein.
it is essentially a set of instructions for how to perform the process, although it may not be phrased exactly that way.
you’re admitting that it’s a pedagogical gimmick and a practical handwave. handwaving is often useful; my criticism is that it is not rigorous, and this is not necessarily a criticism that applies to mathematics proper2, but to those who believe all of accepted mathematics is rigorous, and would attempt to carry over certain handwavy concepts to other fields to “prove” things there.
2it is fine for math to be only as rigorous as it needs to be for the given application…but that last part is key