What makes you think that when I said “All numbers are shapeless” I actually said “All written symbols that we use to represent quantities are shapeless”?
That’s all I need to show you that you’re wrong: tell you that what I said is not what you think I said.
But since you love conventions, I’ll add that if you go to Wikipedia you can see that they make a very clear distinction between numbers and numerals.
Is it so strange that the word “number” most commonly refers to actual numerical quantities and not to symbols (written or spoken) that refer to numerical quantities?
When I said “All horses are numbers” I did not mean “All horses (as conventionally defined) are numbers”. What I meant is “All horses (as I defined the term here in this post) are numbers”.
I did not count how many times you misinterpreted the simplest of my sentences but I believe it’s a lot of times. This goes against your claim that you understand my points too easily. You don’t.
The point of that post of mine (where I spoke of how horses can be used to represent numbers) was to show how changes in definitions can lead to logically invalid arguments. I wasn’t discussing conventions.
Being an asshole is a common sign of having an over-sensitive ego. You are that kind of guy. The overly sensitive guy who has all kinds of silly rationalizations for his irrational behavior. But once again, that’s not the subject of this thread, so let me keep this nice and short.
Nope. Not even close. Dasein is an existential contraption relevant only in regard to moral and political value judgments in the is/ought world. Again, if I do say so myself.
Mathematics [I presume] is anchored wholly in the either/or world.
And, that being case, I presume further that there is in fact one correct answer. But the problem here for mathematically challenged dummies like me, is that I can’t connect the dots between the arguments and the world of actual material/phenomenal interactions.
Some interesting stats. (16%) of Silhouette’s response to the question “Can you give me an example of a number that has a shape?” deals with the actual question and (84%) of it is about who he is. There are (4,601) characters in total.
Guess which subject is more interesting to him. The subject of who he is.
Mathematics, especially modern math, is highly abstract and not bound by any physical considerations. Nobody is saying – well at least I’m not saying – that .999… refers to anything in the real world. On the contrary, I maintain that it does not. Nor does 1! The fact that .999… = 1 is a formal exercise in pure math; and has no point of contact with the real world as described by contemporary physics.
Physics (\neq) Math.
Now of course you will say, well isn’t math used in physics and biology and economics and everyday life at the grocery store and so forth.
Yes. And this is a philosophical mystery. Why is it that math, which is ethereal, way out there, and free of any “ontological burden” to be about the real world; nevertheless turns out to be supremely useful, in fact essential.
The title says it all. Math is so abstract that it is unreasonable to think it applies the world; but it does.
So the best thing to do is remember that math has nothing to do with anything except itself. If others find it useful. the mathematicians are happy but that’s not their reason for doing math. Of course I’m exaggerating a position but I hope I’m making my point.
It’s the job of philosophy to explain why math, a purely conceptual enterprise, is useful at all. Much has been written.
Historically, people used to believe that math described the real world, and that math = physics.
The split between math and physics was caused by the advent of non-Euclidean geometry in the 1840’s. That’s when everyone realized that math could not provide certainty about the world; and that if math couldn’t, then NOTHING could.
That was the birth of postmodernism and the cultural relativism of contemporary society. If rationality can’t tell us what’s true, then maybe rationality’s just a tool of oppression and not a path to truth at all. This all started with non-Euclidean geometry.
Look up the field axioms for the real numbers and it’s resolved. Addition is a binary operation defined on a set. It inputs two elements of the set and outputs a third. We only define x + y for real numbers x and y. By induction we can extend the definition to arbitrary finite sums. But there is NO definition or meaning to an infinite sum.
As I recall Wiki doesn’t have a very good article about this but I found another writeup. Note that addition is a function that inputs two variables and outputs a third. Under this system infinite sums don’t exist or have meaning. We have to define their meaning.
You are historically correct. Infinite sums were used by Newton in the 17th century 200 years before the modern theory of convergence was developed.
But even then, when “everyone knew” what infinite sums were, .999… was still 1. There was never a historical time when it was anything else. All that changed was that we figured out how to prove it logically from first principles.
I understand very well that addition is a function that inputs two numbers and outputs a third number.
However, I don’t see how such a definition (together with other relevant definitions) isn’t enough for us to logically derive the meaning of the expression that is (0.9 + 0.09 + 0.009 + \cdots).
Your conclusion (that this isn’t impossible) looks like a non-sequitur.
I believe you will have to make your argument a bit more explicit because as it is it is very hard, if not impossible, to evaluate its validity.
How does the conclusion (that addition is not defined for infinite sums) follow from the premise (that addition is a function that takes two numbers and returns a single number)? Where’s the link?
Hopefully, you are NOT saying that just because addition is a function that takes TWO inputs and not AN INFINITE NUMBER of inputs it follows that addition is not defined for infinite sums. Using that kind of logic, one can also conclude that addition is not defined for any sum that involves any number of terms other than (2) which you agree is not the case.
The point was to make my position clear and visible so that other people can address it and not address a strawman.
Did I explicitly state that you or anyone else is stupid? I don’t think so. Indeed, I don’t know what would be the point of doing that. To disagree with someone is not to say that they are stupid.
Yes, I do have eyes, and yes, there is a reason you singled me out to be impatient with. However, what you think is the reason you singled me out is something that can be disputed.
By the way, note that I did not say that you are impatient ALL OF THE TIME and with EVERY person who disagrees with you.
The way I see it, you have no problem with people who disagree with you so as long they do not threaten your ego or otherwise frustrate you. Is that right?
I don’t think I present myself as “indisputably correct”. What I did here in this thread is 1) I presented my opinions and 2) I did not change them in response to what you and other people had to say. This does not mean I am correct, of course, it can simply mean that I fail to see the point that other people are making despite their efforts. But also, it can mean that none of you have any arguments whatsoever.
I don’t think the things we’re discussing here are advanced and insightful. And whether you or anyone else thinks the subject of this thread is advanced and insightful is, to be perfectly honest with you, completely irrelevant.
Note that these kinds of things are typically said by people who are insane.
Maybe you don’t see the point in continuing this discussion but I do.
Consider a word such as “bow”. The word “bow” has multiple meanings. Here are two of them:
a long wooden stick with horse hair that is used to play certain string instruments such as the violin
a weapon to shoot projectiles with (e.g. a bow and arrow)
Here’s a question:
Does this mean that a long wooden stick that is used to play certain string instruments such as the violin is the same thing as a weapon used to shoot projectiles with?
Of course not. They are two different things.
Similarly, (f(x) = 0) might have more than one meaning. For example:
a function that maps every natural number to (0)
(0)
Does that mean that a function that maps every natural number to (0) is the same thing as (0)?
Of course not. They are two different things.
In one sense, functions are not numbers. In another sense, functions are numbers. In this other sense, the word “function” means no more than “number”. This is not what the word “function” normally means and what it originally meant. It’s a subsequent development. And this kind of thing (taking existing words that have an established meaning and then giving them a different meaning) can lead to equivocation.
So, aside from it being a “philosophical mystery”, I’m not entirely sure what you mean here by “no connection”. But, as you note, out in the either/or world there seem to be countless connections.
Though I suspect that here we may be getting closer to the points that folks like Wittgenstein raised. The relationship between words and worlds. Language itself.
Besides, just on a day to day practical level, it’s important to accept that if John owns two horses and buys two more, he now has four horses and not five.
But 1 horse vs. .999… horse? That’s the part I can’t wrap my head around. Though I am perfectly willing to acknowledge that this revolves far more around my ignorance in regard to mathematics on this level.
No, where dasein comes in for me is when someone will insist that no one should be permitted to buy and sell horses. Period. That owning animals is immoral.
Indeed, I know someone who insists that this includes cats and dogs.
For purposes of discussion, suppose I grant you that point. Then show me any such logical derivation.
Previously you’ve said that no definition is needed because the meaning is known to everyone or obvious or intuitive or whatever, I apologize if I’m not quoting you perfectly but this is the feeling I get.
But now you are saying is that yes, you AGREE that infinite sums aren’t defined UNTIL we LOGICALLY DERIVE their meaning.
So, I do ask you to simply outline such a logical derivation.
What do you think modern real analysis is, other than a logical derivation from first principles of what limits and infinite sums are?
Addition is defined ONLY for finite sums. One of the axioms for the real numbers says that:
The sum of any two real numbers is a real number. In symbols, if (x) and (y) are real numbers, then so is (x + y).
So when we see an expression, or string of symbols such as (x_1 + x_2 + x_3 + \dots), we have no rules with which to untangle or parse that string. If it’s the sum of 47 elements, it’s the sum of the first 46 plus the 47th, and that’s an instance of adding only two things. So by induction, any finite number of additions are possible.
But the theory says nothing about infinite sum. No meaning is ascribed to them. We need additional ideas and formalizations, namely the modern theory of real analysis.
Of course if you have a different way of doing it, I’m all ears. I have said that I am openminded. I’m not defending mathematical orthodoxy, only describing it. If you have another way to define infinite sums, let’s hear it.
If you have a vending machine that lets you put in two coins, you’ll break it if you try to stuff in infinitely many coins.
LOL. But that’s exactly what I’m saying. And if you disagree, then tell me what is the definition of an infinite sum, based on the axiom that the sum of any TWO real numbers is a real number?
A simple inductive argument serves to generalize addition to any finite number of summands, as I outlined a couple of paragraphs ago.
It’s very much like parsing a computer language. If we see x + y + x + w we know we can parenthesize (x + y + z) + w to reduce the problem to a sum of two things, which we have a procedure for.
But if we have an infinite sum, the procedure never terminates. We do not have a computable function that allows us to determine the sum of infinitely many terms.
Do you see that? A machine could compute 1/2 + 1/4 + 1/8 by continually breaking it down into smaller sums, a process that is guaranteed to terminate. But if the input is an infinite sum, the evaluation never terminates and no such sum can be reached. We do not have a procedure or algorithm to calculate an infinite sum that is guaranteed to terminate after finitely many steps.
That’s why we need a new idea. The idea of limits. The theory of limits goes beyond the theory of computation, you can look at it that way. Indeed, the difference between arithmetic and analysis is exactly that. Arithmetic is everything you can do with finite operations. Analysis deals with infinitary operations.
In fact, finite addition can be defined in Peano arithmetic. The theory of limits requires the axiom of infinity.
You are failing to appreciate this conceptual leap. Finite sums do not require the axiom of infinity. Infinite sums do.
Then by all means restate your thesis; rather than provide argument or evidence.
That’s not an example of anything. The integers are a ring. In a ring you can add, subtract, and multiply, but not necessarily divide.
If you enlarge the ring to a field, then you can always divide (except by 0). This is perfectly well understood. Your post was a little off the mark IMO.
Now if one is interested in integer division, we know that 5/2 is 2R1; that is, quotient 2 and remainder 1. That’s the domain of number theory.
You seem to be going off into unproductive ideas about this.
Oh yes of course. Better if I’d phrased it “not necessarily any known connection.” I only meant that people trying to understand .999… should not try to connect it with physical reality, since in general math isn’t. I didn’t mean to discount the entire philosophy of the subject. That was the philosophical mystery part.
I don’t know much about his work, and what little I’ve read I didn’t understand. I’ve heard that he was wrong about a couple of mathematical things.
I am a Platonist during the week and a Formalist on weekends. I did not mean to deny the “unreasonable effectiveness,” nor the fact that mathematics came out of the markets of the ancient world and seems like part of nature, or at least part of our own nature.
Oh I see. In connection with this thread I’ve been reviewing the history of infinite sums. Newton was proficient at using infinite sums to solve problems in physics and math. Infinite sums and infinitary processes came into math and science around that time. But you can’t apply the real numbers to physical measurements, that’s well known. So nobody is saying that .999… horses is one horse. That’s not true. What’s true is that the real number .999… and the real number 1 are the same real number.
You’ve touched on the question of whether the integer 1 is the same thing as the real number 1. At best they seem to be two aspects of the same thing that are used for different purposes. Yet their nature is very different, as you note. The discrete and the continuous. Back to Democritus.
They are exactly the same thing. The function that is identically zero is often taken as the zero element in a ring of functions. Would you call that a number or not call it a number? In math there is no universal definition of number!! So you really can’t say.
Zero is just the zero element in a group or ring. Whether it’s the zero function in the ring of continuous functions; or the zero of the integers; it’s just zero. In Platonic heaven there is only one zero, and it has many guises. Zero is not a thing. Zero is characterized by what it does. And what it does is act as the additive identity for any system that has addition and an additive identity! It’s the thing that if you add it to another thing, the output is the other thing.
Nobody knows what zero “is.” We only know what zero does. If a thing acts like zero, it’s zero. Both the zero of the integers and the function that’s identically zero in the ring of continuous functions, act like zero. So they’re zero.
This by the way is a vague hint of mathematical structuralism: the idea that mathematical objects aren’t characterized by what they are; but rather by what they do, and how they stand in relation to other objects.
The meaning of a sentence is logically derived from the meaning of the individual terms that constitute it.
For example, the meaning of the statement “All bachelors are married” is logically derived from the meaning of the individual terms that consistute it and these are “all”, “bachelor”, “are” and “married”. In other words, you derive its meaning, you do not freely invent a new one e.g. you do not just freely declare that what the statement refers to is a cat.
In the same exact way, the meaning of the expression that is (0.9 + 0.09 + 0.009 + \cdots) is logically derived from the terms that constitute it and these are (+), (\dotso) and numbers of the form (9 \times 10^{-n}) where (n) is a natural number. In other words, you derive its meaning, you do not freely invent a new one e.g. you do not freely declare that what it represents is the limit of the sum that is (0.9 + 0.09 + 0.009 + \cdots).
You do understand that this axiom says NOTHING other than that the sum of TWO real numbers is a real number? Hell, it does not even tell us the actual result of every (or any) two real numbers, or at least, how to calculate the result of any two real numbers. It merely tells us that the result of every two real numbers belongs to the category of numbers that is “real numbers”.
I agree. The same goes for an infinite number of additions.
I disagree. Note that you didn’t explain WHY. You merely asserted that we have no rules with which to untagle or parse that string.
But you’d break it by trying to stuff in ANY number of coins other than two. The vending machine lets you put no more than two coins at the same time. So if you want insert more than two coins at the same time, there is no way to do it. But you can insert ANY number of coins (even an infinite number of them) provided you don’t try to insert them simultaneously.
I agree.
That’s true.
However, I am afraid this is irrelevant. We shall see why shortly.
That’s correct. A machine cannot calculate the result of an infinite sum by iterating through an infinite array of terms that constitute it.
But that is a completely irrelevant insight.
There are OTHER ways to figure out the result of an infinite sum. For example, an infinite sum of zeroes is zero. We can know this because zero times infinity is zero (or more generally, because zero times any quantity is zero.) It doesn’t matter how many (0)'s you add, the result will always be (0). No infinite loop is required in order to calculate the result.
In the same exact way, one can know that (0.9 + 0.09 + 0.009 + \cdots) is greater than (0.9) as well as less than (1). No infinite loop required.
One step at a time. People must first understand what is it that I’m saying (i.e. what is it that I have to prove.)
I can agree that there is a sense in which it is correct to say that functions are numbers (e.g. if you redefine the word “function” to literally mean “number”) but to say this is the original sense, or even worse, to say there is no other sense in which functions are not numbers is something I do not agree with.
I am not sure about others but I know very well what it means to have “zero money” i.e. what “zero money” is. On the other hand, I have absolutely no idea what it means to say that “zero money” is doing something.
That’s understandable.
However, to claim that the concept of zero is dependent on the concept of addition (or any other operation), in the sense that one cannot know what zero is without knowing what addition (or some other operation) is, is taking it too far. That would be muddying the waters . . .
I can show you a static picture such as this one and tell you “There are zero dragons in it” without knowing anything about mathematical operations.
The concept of function and the concept of number are two INDEPENDENT concepts. You can’t reduce one to another, and certainly, you can’t say they are synonyms.
What if the concept “(1)” is constructed to signify the quantity of horses that you’re seeing, and the concept “(0.\dot9)” is also constructed to signify that same quantity of horses that you’re seeing?
Either way, the sensory data you’re receiving indicates what you know to call “(1) horse”, so if some other way of representing that same quantity of horses is “(0.\dot9) horse”, what’s there to wrap your head around?
Just because the representations look different, and maybe you associate anything that looks like 0.9 with less than 1, does the horse look any different? The two representations of the number of horses are constructed to signify the same quantity, so it shouldn’t.
In mathematics, unlike other subjects for which non-mathematicians have much better intuitions, the necessary amount of precision required allows for exactly correct answers.
Fortunately, you have the right attitude, approaching a subject that you regard yourself to be ignorant in with honesty and humility, admitting it’s simply something that you struggle to get your head around rather than asserting with unshakable conviction that an understanding known to be false is true just because you don’t understand why, which puts you in a position where you could potentially learn if you’re interested. Don’t you agree that such an approach is the only way to approach philosophy and conduct a productive discussion?
The alternative is apparently to be told why mathematical knowledge is what it is, and to conclude as a non-mathematician that mathematics is wrong regardless.
It’s not the case that mathematical knowledge is what it is “because mathematics says so”, the whole point is that anyone can construct it from the bare bones of logic and arrive at the same conclusions if they’re able and don’t suffer from biases and other hindrances to objective thinking. That’s why attitude and approach are so fundamentally important, even before mathematical discussion can even get off the ground. Evidently it’s a mistake to even begin to engage with anyone on a mathematical level when they lack the most basic ability to think objectively and approach argument with honesty and humility (and not simply “claim” you are, but actually doing so), regardless of their mathematical background or aptitude.
The starting position is irrelevant. In the case of iambiguous, the starting position is that of ignorance i.e. he has no opinion on the matter. You can’t expect people to be EITHER right OR neutral. Most people are opinionated. The most you can hope is that they are willing to EXAMINE their reasoning for potential flaws. (And if they are not, all you have to do is adjust your expectations and act accordingly. Nothing to be frustrated with.)
I agree that (1 = 0.5 + 0.5) and that (1 = 0.25 + 0.25 + 0.25 + 0.25). (Your wording is a bit off. Number (1) divided in half is (0.5), it is not (0.5 + 0.5).) But how does it logically follow that every “fucking” number equals zero?