Don't know if this is the right place, but I have a question for anyone who knows something about probability, math, computer science and maybe philosophy of modality:
If I have a random, countably infinite, set containing unique values, say the set of positive integers, what is the probability that if I were to pull a number from the set that that number would be 823? Would it be 0 or would it be indeterminate? I can't wrap my mind around this and need an explanation. Also, would this be the same for uncountably infinite set?
In rough english:
If I happened to have a really large bag containing the numbers 1 to infinity and decided to pull one number out of the bag at random, what's the chance that I would pull out the number 41? Is it 0% or is it impossible for the probability to be calculated?
infinite is not a real number, more of a concept really.
To answer what you asked, the percentage points of drawing it are 0 for all practical uses. This is referred to as an infinitesimal. Infinitesimals are used to express the idea that objects so small have no way to simple way see them or to measure them. Another one is: .999.... = 1. The difference between the 2 numbers is so small, that we assume there is none.
EDS: Before an argument over my example arises, there is proof.
1/3= .333....
3 x 1/3 = 3 x .333....
1 = .999.....
The Kolmogorov axioms for probability, which require all probabilities to be real numbers, the probability measure to be countably additive, and the integral of the measure across the entire probability space be equal to 1, imply that the probability of events of the sort that you describe are undefined.
This is because if you assign any real probability p > 0 to each integer, then the total measure of the probability space becomes (by countable additivity) lim(n->inf) n*p = inf, a violation of the axiom that the total measure equals one. Similarly, if you set p = 0, you get zero for the total measure.
So there is no real number you can assign. But since you had to assign a real number, you're forced to acknowledge the thing you were looking for can't exist.
This is because there is no countably additive real-valued uniform measure over the integers. Now, if you are willing to slightly bend the rules of probability and allow non-real-valued measures -- let's say the surreal numbers -- then you can give each integer a probability of 1/omega. Another thing you can do is choose a non-uniform distribution, that is to say, you can choose any function whose integral from 1 to infinity is equal to one and use that as the probability density for picking a given integer. But that seems to distort the intent of what you're asking.
The Kolmogorov axioms for probability, which require all probabilities to be real numbers, the probability measure to be countably additive, and the integral of the measure across the entire probability space be equal to 1, imply that the probability of events of the sort that you describe are undefined.
This is because if you assign any real probability p > 0 to each integer, then the total measure of the probability space becomes (by countable additivity) lim(n->inf) n*p = inf, a violation of the axiom that the total measure equals one. Similarly, if you set p = 0, you get zero for the total measure.
So there is no real number you can assign. But since you had to assign a real number, you're forced to acknowledge the thing you were looking for can't exist.
This is because there is no countably additive real-valued uniform measure over the integers. Now, if you are willing to slightly bend the rules of probability and allow non-real-valued measures -- let's say the surreal numbers -- then you can give each integer a probability of 1/omega. Another thing you can do is choose a non-uniform distribution, that is to say, you can choose any function whose integral from 1 to infinity is equal to one and use that as the probability density for picking a given integer. But that seems to distort the intent of what you're asking.
This is likely the well-defined answer I was looking for. Thanks! And yeah, I should've mentioned that the weights for each element in the set were equal, i.e. a uniform distribution.
If I happened to have a really large bag containing the numbers 1 to infinity and decided to pull one number out of the bag at random, what's the chance that I would pull out the number 41? Is it 0% or is it impossible for the probability to be calculated?
you can express that probability as a limit.
P = probability of pulling 41 from the set
n = size of set
1 = number of draws
P = 1/n
the limit of P as n approaches infinity IS zero. This is not to say that 1/infinity is equal to zero, that's just the limit. For all intents and purposes, zero can be used in real-life applications well enough. I would say that the probability is zero. However: where are you going to find an infinite set in the real world? even the number of hydrogen atoms in the sun is a finite quantity.
math is a tool to examine the real world. people often forget that, and think that reality is based off of math (as in a computer simulation). If your mathematical constructs do not have real-life analogs/equivalents, you're doing math wrong.
math is a tool to examine the real world. people often forget that, and think that reality is based off of math (as in a computer simulation). If your mathematical constructs do not have real-life analogs/equivalents, you're doing math wrong.
This type of math has practical application.
The question went over everybody's head but Crashing00, who I believe answered I correctly. I understood the question, but admit I do not remember the explanation for the answer, since it's been about 30 years since I studied this stuff.
infinite is not a real number, more of a concept really.
To answer what you asked, the percentage points of drawing it are 0 for all practical uses. This is referred to as an infinitesimal. Infinitesimals are used to express the idea that objects so small have no way to simple way see them or to measure them. Another one is: .999.... = 1. The difference between the 2 numbers is so small, that we assume there is none.
EDS: Before an argument over my example arises, there is proof.
1/3= .333....
3 x 1/3 = 3 x .333....
1 = .999.....
Without a clear definition of what you mean by a "number," it's hard to say whether or not infinity is a number. Certainly there's many common standards of "number" that infinity would adhere to.*
Further, I think you're a bit confused about infinitesimals and the equivalence of .999... = 1. In fact, under standard analysis your definition of the infinitesimal would prevent the equality!
The fractional proof of 1 = .999... is fallacious, you're assuming the conclusion in the first line of the proof.
*I am assuming outright that when you said "a real number" you did not mean "a member of R"
If your mathematical constructs do not have real-life analogs/equivalents, you're doing math wrong.
If you're going to follow this principle you'll end up throwing out a huge amount of modern pure math.
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EDS: Before an argument over my example arises, there is proof.
1/3= .333....
3 x 1/3 = 3 x .333....
1 = .999.....
That's sketchy. The better way is to represent 0.99999... as the sum of 9/(10^x) for x = 1,2,3,... (an infinite series). Then by a theorem the sum is equal to 9/(10-1) = 9/9 = 1.
If your mathematical constructs do not have real-life analogs/equivalents, you're doing math wrong.
No, it's perfectly fine to go abstract, and there are many reasons to do so. Firstly, it helps refine the methodology, so you might be able to build better (more precise) models. It could also lead to new and surprising conclusions, so you can model things that were previously impossible. At its highest level, you get to appreciate the elegant complexity of mathematics as a logical structure: "Look at all this stuff that follows from just these few laws of logic and these few axioms!"
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math is a tool to examine the real world. people often forget that, and think that reality is based off of math (as in a computer simulation). If your mathematical constructs do not have real-life analogs/equivalents, you're doing math wrong.
Math is a tool to examine the real world, yes. And like any tool, it can be applied in any way you choose to apply it. (You don't ask for a real-world analog/equivalent for a hammer.) Exploring the full functionality of the tool in the abstract is an excellent way of finding applications you didn't even know existed. There have been many, many cases in science of researchers discovering that a real-world phenomenon is isomorphic to a mathematical function or principle or technique.
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candidus inperti; si nil, his utere mecum.
rld? even the number of hydrogen atoms in the sun is a finite quantity.
math is a tool to examine the real world. people often forget that, and think that reality is based off of math (as in a computer simulation). If your mathematical constructs do not have real-life analogs/equivalents, you're doing math wrong.
Reality is not based off of math, reality is its own thing, and math is simply the patterns we observe in it put into numeric values.
Reality is not based off of math, reality is its own thing, and math is simply the patterns we observe in it put into numeric values.
Mathematic structures extend far beyond what we observe in the natural world, and they do seem to serve more than just a descriptive quality- at the very least, it's certainly true that we analyze mathematic descriptions beyond the immediate and use this to generalize scientific hypotheses that prove to be true decades later.
There's also idea like that Mathematical Universe hypothesis and Grand Ensemble theory that suppose the whole universe is, first and foremost, a mathematic construct.
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Mathematic structures extend far beyond what we observe in the natural world, and they do seem to serve more than just a descriptive quality- at the very least, it's certainly true that we analyze mathematic descriptions beyond the immediate and use this to generalize scientific hypotheses that prove to be true decades later.
There's also idea like that Mathematical Universe hypothesis and Grand Ensemble theory that suppose the whole universe is, first and foremost, a mathematic construct.
If you think math is an existent thing on its own, and not just something we made up to fit patterns we see, show me a natural number 2 just floating in space.
If you think math is an existent thing on its own, and not just something we made up to fit patterns we see, show me a natural number 2 just floating in space.
Two issues here:
1. You've created a false dichotomy. Basic logic, for instance, is certainly true regardless of whether or not there's minds to perceive it. There's no empirical quality to logic. In a more nebulous sense this is true of maths as well, there's vast branches of mathematic research that have absolutely no real-world correspondence.
2. There's a variety of problems with requiring that there must be a "natural number 2 just floating in space." Firstly, there are certainly real properties of things that nonetheless don't exist independently of material objects- I'm sure you concede that gravity or oscillations are not just illusory, for instance, despite them only existing in reference to other objects. The second is that even the very extreme forms of mathematical realism (like the mathematical universe hypothesis I mentioned) wouldn't require conscious observes to be able to view small mathematic structures like the Naturals, they believe that the universe itself is a grand mathematic structure and that all mathematic structures can be said to be real- not that all mathematic structurse are embdedded as free-floating entities in our own universe.
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Two issues here:
1. You've created a false dichotomy. Basic logic, for instance, is certainly true regardless of whether or not there's minds to perceive it. There's no empirical quality to logic. In a more nebulous sense this is true of maths as well, there's vast branches of mathematic research that have absolutely no real-world correspondence.
2. There's a variety of problems with requiring that there must be a "natural number 2 just floating in space." Firstly, there are certainly real properties of things that nonetheless don't exist independently of material objects- I'm sure you concede that gravity or oscillations are not just illusory, for instance, despite them only existing in reference to other objects. The second is that even the very extreme forms of mathematical realism (like the mathematical universe hypothesis I mentioned) wouldn't require conscious observes to be able to view small mathematic structures like the Naturals, they believe that the universe itself is a grand mathematic structure and that all mathematic structures can be said to be real- not that all mathematic structurse are embdedded as free-floating entities in our own universe.
All theories of gravity are based off of real, non-illusionary objects. The fabric of space is said to be a real thing, and virtual particles are said to exist. Mathematics did not exist before humanity as far as we know, only the universe did.
I can use basic logic to prove what I am saying. The universe is the universe. Do you disagree? The universe is not math, it's the universe. Just because I bring two apples close together doesn't mean there is some mystical force adding them. In reality, they are simply two apples that happen to be in close proximity to each other, and labeling them as a group is something we made up. Perhaps there are patterns with objects that happen to work in seemingly quantitative ways, but those objects themselves are not math.
All theories of gravity are based off of real, non-illusionary objects. The fabric of space is said to be a real thing, and virtual particles are said to exist. Mathematics did not exist before humanity as far as we know, only the universe did.
I can use basic logic to prove what I am saying. The universe is the universe. Do you disagree? The universe is not math, it's the universe. Just because I bring two apples close together doesn't mean there is some mystical force adding them. In reality, they are simply two apples that happen to be in close proximity to each other, and labeling them as a group is something we made up. Perhaps there are patterns with objects that happen to work in seemingly quantitative ways, but those objects themselves are not math.
I'm not trying to be rude, but can I have some idea as to your background? It doesn't affect the validity of what you say or make you inferior in anyway, it's just that I think I might be assuming shared familiarity with some foundational ideas and misleading you as a result.
Edit: If you don't wish to share of find it to be off topic, I'd be happy to continue on as is.
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I'm not trying to be rude, but can I have some idea as to your background? It doesn't affect the validity of what you say or make you inferior in anyway, it's just that I think I might be assuming shared familiarity with some foundational ideas and misleading you as a result.
Edit: If you don't wish to share of find it to be off topic, I'd be happy to continue on as is.
I do think it is somewhat off topic but I do know calculus and studied some number theory, it literally took me two pages to explain why 1+2=3 because I had to proof every step and label what 1+2 and 3 were and prove that those labels were accurate. In short I think math is logical, but I don't think it itself is the universe. I think they are the patterns we find in it recorded as symbols on a piece of paper.
Even if you look at the qnautinization of orbitals, you can still explain it qualitatively this way: "The probability of an electron reaches a point below possible measurability in certain areas".
I do think it is somewhat off topic but I do know calculus and studied some number theory, it literally took me two pages to explain why 1+2=3 because I had to proof every step and label what 1+2 and 3 were and prove that those labels were accurate. In short I think math is logical, but I don't think it itself is the universe. I think they are the patterns we find in it recorded as symbols on a piece of paper.
What on earth sort of course would force you to "prove those labels were accurate?"
Did your number theory course include proof-calculus based classes (this is very different from your "calculus" classes and if you're not sure the answer is no) andshould I assume then that you don't have much experience with foundations of mathematics or ontology? Topology?
Edit: again, it's not problematic if you haven't. It just means that I'll be more careful about such topics.
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Did your number theory course include proof-calculus based classes (this is very different from your "calculus" classes and if you're not sure the answer is no) andshould I assume then that you don't have much experience with foundations of mathematics or ontology? Topology?
In calculus you do prove things, but not like in number theory, you do simpler proofs, such as to test if a formula works, like if a formula you wrote is the correct formula for a set of numbers, it has be equal the same answer as what your deriving it from at an input of 1 AND x+1, or always the next number. I think there was some guy who wrote a rule about prime numbers, and he tested it with 1000 prime numbers, but he never tested it with "x+1", and so eventually someone found a prime number that the formula didn't work for. Or you do things like you prove trigonometric functions like proving tan is equal to sin/cos using certain properties and finding derivatives and etc.
Edit: again, it's not problematic if you haven't. It just means that I'll be more careful about such topics.
No, I know what I'm trying to say, but perhaps I'm not saying it right.
A piece of paper is not math, even if you can put math on it, a pencil is not math just because you can right math with it. An atom is not math just because it happens to have patterns that you can happen recognize with numbers that we can write with a pencil on a piece of paper.
If I label 1 as a', all its principals have to hold through. a' has to be a' and the properties of a' being a' cannot change.
I understand what relabeling is, what I'm asking is what sort of course would require you to defend the labeling of 1, 2, 3 and the basic operators of arithmetic.
In calculus you do prove things, but not like in number theory, you do simpler proofs, such as to test if a formula works, such as that the function has to equal the same answer as what your deriving it from at 1 AND x+1, or always the next number.
Proof calculus is not what you did. What you did was differential calculus with proofs. The phrase "____ calculus" is widely used in mathematics to apply to a large variety of fields.
No, I know what I'm trying to say, but perhaps I'm not saying it right.
A piece of paper is not math, even if you can put math on it, a pencil is not math just because you can right math with it. An atom is now math just because it happens to have patterns that you can recognize with numbers.
The issue is not what you are trying to say. The issue is, pardon the assumptions, that you are trying to portray a much broader background than you actually have. The difficulty is that if, assuming good faith, you are responded to with the expectation that you have such a background, problems arise when the people you are speaking to respond with the assumption you know basic principles you have not yet learned.
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I understand what relabeling is, what I'm asking is what sort of course would require you to defend the labeling of 1, 2, 3 and the basic operators of arithmetic.
Number theory would. Perhaps I'm not explaining it right, but a' is a' because of a specific property called the reflexive property which sometimes you have to state. I suppose I did hate number theory a lot and I only passed with a C, but it's hard for me to think I put all that work into it just to be wrong about something like this.
Proof calculus is not what you did. What you did was differential calculus with proofs. The phrase "____ calculus" is widely used in mathematics to apply to a large variety of fields.
Perhaps it is, but just just goes to show number theory and "calculus with proofs" are different.
The issue is not what you are trying to say. The issue is, pardon the assumptions, that you are trying to portray a much broader background than you actually have. The difficulty is that if, assuming good faith, you are responded to with the expectation that you have such a background, problems arise when the people you are speaking to respond with the assumption you know basic principles you have not yet learned.
Ok, so aside from what I'm saying not actually being wrong, we're trying to discuss if every object in the universe is math, how does even having a background in number theory help that when number theory can't prove or disprove that? It's philosophy. Number theory is just the logic to prove things in mathematics. It seems like your trying to discredit my knowledge math so that you can say "He doesn't know math, so his philosophy doesn't work", even though math and philosophy are two completely different unrelated topics. I think at this point it is going off topic.
I attempted to prove it, but then I realized it would take pages even in a simplified way, because I would have to prove not only that .333... is equal to 1/3 and that .999 is equal to 1, but also that .333.... and .999... are cyclic numbers.
I suppose if I said
x was equal to .333 then multiplied each line by 10, so
x=.333...
10x=3.333...
then subtracted the first line from the second line to make 9x=3,
10x=3.333...
x=0.333...
-
__________
9x=3
then divided x and 3 by 9 to make x=3/9 I could say 3/9=1/3 and then 1/3=x=.333 and 1/3 = .333 via the transitive property and use that to say that 3 x .333... = 3 x 1/3 = 1 = .999.... and I could probably get away with transitive property again on this forum.
Actually proving it with number theory though would take a lot of work, something I'm not willing to do for some forum post and neither is Macius probably.
I attempted to prove it, but then I realized it would take pages even in a simplified way, because I would have to prove not only that .333... is equal to 1/3 and that .999 is equal to 1, but also that .333.... and .999... are cyclic numbers.
I suppose if I said
x was equal to .333 then multiplied each line by 10, so
x=.333...
10x=3.333...
then subtracted the first line from the second line to make 9x=3,
10x=3.333...
x=0.333...
-
__________
9x=3
then divided x and 3 by 9 to make x=3/9 I could say 3/9=1/3 and then 1/3=x=.333 and 1/3 = .333 via the transitive property and use that to say that 3 x .333... = 3 x 1/3 = 1 = .999.... and I could probably get away with transitive property again on this forum.
Actually proving it with number theory though would take a lot of work, something I'm not willing to do for some forum post and neither is Macius probably.
I don't think you know what number theory is. It would not include proofs of real number equality. I have stopped responding to your posts because it has gotten exhausting, but please stop posturing.
Also, your proof is invalid.
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I don't think you know what number theory is. It would not include proofs of real number equality. I have stopped responding to your posts because it has gotten exhausting, but please stop posturing.
Um, yes it can, but your always given some axioms but not always the same ones, and sometimes those axioms are "number ABC is a real number" or "XYZ is divisible by three" which you can use to say "If the real number XYZ is divisible by three, every permutation of XYZ is divisible by three." But let's say your proving a geometric theorem where two shapes share the same side, at one point you can say A of triangle BAC = A of triangle DAF; reflexive property" which is where you'd state A=A.
Just as a passive observer here, FakeMcCoy, what are you talking about? Did you really take Number Theory in college? I did, and you just don't sound like you know anything about it other than some problems you dabbled with. It really sounds like you don't know what you're talking about. You don't sound dumb, actually you sound smart... but you sound like you're over-representing your knowledge base.
I'm not necessarily siding with paths... but you drop gads of jargon in a way that makes it sound like you're hinting you have some deep understanding of the math.
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If I have a random, countably infinite, set containing unique values, say the set of positive integers, what is the probability that if I were to pull a number from the set that that number would be 823? Would it be 0 or would it be indeterminate? I can't wrap my mind around this and need an explanation. Also, would this be the same for uncountably infinite set?
In rough english:
If I happened to have a really large bag containing the numbers 1 to infinity and decided to pull one number out of the bag at random, what's the chance that I would pull out the number 41? Is it 0% or is it impossible for the probability to be calculated?
Also, I found this article, though slightly irrelevant, really interesting: http://plato.stanford.edu/entries/chance-randomness/
To answer what you asked, the percentage points of drawing it are 0 for all practical uses. This is referred to as an infinitesimal. Infinitesimals are used to express the idea that objects so small have no way to simple way see them or to measure them. Another one is: .999.... = 1. The difference between the 2 numbers is so small, that we assume there is none.
EDS: Before an argument over my example arises, there is proof.
1/3= .333....
3 x 1/3 = 3 x .333....
1 = .999.....
540 Peasant cube- Gold EditionSomething SpicyThis is because if you assign any real probability p > 0 to each integer, then the total measure of the probability space becomes (by countable additivity) lim(n->inf) n*p = inf, a violation of the axiom that the total measure equals one. Similarly, if you set p = 0, you get zero for the total measure.
So there is no real number you can assign. But since you had to assign a real number, you're forced to acknowledge the thing you were looking for can't exist.
This is because there is no countably additive real-valued uniform measure over the integers. Now, if you are willing to slightly bend the rules of probability and allow non-real-valued measures -- let's say the surreal numbers -- then you can give each integer a probability of 1/omega. Another thing you can do is choose a non-uniform distribution, that is to say, you can choose any function whose integral from 1 to infinity is equal to one and use that as the probability density for picking a given integer. But that seems to distort the intent of what you're asking.
Which if thou dost not use for clearing away the clouds from thy mind
It will go and thou wilt go, never to return.
Hey! Hey! Didn't I say rough English? How am I supposed to put a problem in layman's terms if I have to mention that infinity is not a real number?
This is likely the well-defined answer I was looking for. Thanks! And yeah, I should've mentioned that the weights for each element in the set were equal, i.e. a uniform distribution.
you can express that probability as a limit.
P = probability of pulling 41 from the set
n = size of set
1 = number of draws
P = 1/n
the limit of P as n approaches infinity IS zero. This is not to say that 1/infinity is equal to zero, that's just the limit. For all intents and purposes, zero can be used in real-life applications well enough. I would say that the probability is zero. However: where are you going to find an infinite set in the real world? even the number of hydrogen atoms in the sun is a finite quantity.
math is a tool to examine the real world. people often forget that, and think that reality is based off of math (as in a computer simulation). If your mathematical constructs do not have real-life analogs/equivalents, you're doing math wrong.
The question went over everybody's head but Crashing00, who I believe answered I correctly. I understood the question, but admit I do not remember the explanation for the answer, since it's been about 30 years since I studied this stuff.
Without a clear definition of what you mean by a "number," it's hard to say whether or not infinity is a number. Certainly there's many common standards of "number" that infinity would adhere to.*
Further, I think you're a bit confused about infinitesimals and the equivalence of .999... = 1. In fact, under standard analysis your definition of the infinitesimal would prevent the equality!
The fractional proof of 1 = .999... is fallacious, you're assuming the conclusion in the first line of the proof.
*I am assuming outright that when you said "a real number" you did not mean "a member of R"
If you're going to follow this principle you'll end up throwing out a huge amount of modern pure math.
That's sketchy. The better way is to represent 0.99999... as the sum of 9/(10^x) for x = 1,2,3,... (an infinite series). Then by a theorem the sum is equal to 9/(10-1) = 9/9 = 1.
No, it's perfectly fine to go abstract, and there are many reasons to do so. Firstly, it helps refine the methodology, so you might be able to build better (more precise) models. It could also lead to new and surprising conclusions, so you can model things that were previously impossible. At its highest level, you get to appreciate the elegant complexity of mathematics as a logical structure: "Look at all this stuff that follows from just these few laws of logic and these few axioms!"
Very Well Then I Contradict Myself.
Math is a tool to examine the real world, yes. And like any tool, it can be applied in any way you choose to apply it. (You don't ask for a real-world analog/equivalent for a hammer.) Exploring the full functionality of the tool in the abstract is an excellent way of finding applications you didn't even know existed. There have been many, many cases in science of researchers discovering that a real-world phenomenon is isomorphic to a mathematical function or principle or technique.
candidus inperti; si nil, his utere mecum.
Reality is not based off of math, reality is its own thing, and math is simply the patterns we observe in it put into numeric values.
Mathematic structures extend far beyond what we observe in the natural world, and they do seem to serve more than just a descriptive quality- at the very least, it's certainly true that we analyze mathematic descriptions beyond the immediate and use this to generalize scientific hypotheses that prove to be true decades later.
There's also idea like that Mathematical Universe hypothesis and Grand Ensemble theory that suppose the whole universe is, first and foremost, a mathematic construct.
If you think math is an existent thing on its own, and not just something we made up to fit patterns we see, show me a natural number 2 just floating in space.
Two issues here:
1. You've created a false dichotomy. Basic logic, for instance, is certainly true regardless of whether or not there's minds to perceive it. There's no empirical quality to logic. In a more nebulous sense this is true of maths as well, there's vast branches of mathematic research that have absolutely no real-world correspondence.
2. There's a variety of problems with requiring that there must be a "natural number 2 just floating in space." Firstly, there are certainly real properties of things that nonetheless don't exist independently of material objects- I'm sure you concede that gravity or oscillations are not just illusory, for instance, despite them only existing in reference to other objects. The second is that even the very extreme forms of mathematical realism (like the mathematical universe hypothesis I mentioned) wouldn't require conscious observes to be able to view small mathematic structures like the Naturals, they believe that the universe itself is a grand mathematic structure and that all mathematic structures can be said to be real- not that all mathematic structurse are embdedded as free-floating entities in our own universe.
All theories of gravity are based off of real, non-illusionary objects. The fabric of space is said to be a real thing, and virtual particles are said to exist. Mathematics did not exist before humanity as far as we know, only the universe did.
I can use basic logic to prove what I am saying. The universe is the universe. Do you disagree? The universe is not math, it's the universe. Just because I bring two apples close together doesn't mean there is some mystical force adding them. In reality, they are simply two apples that happen to be in close proximity to each other, and labeling them as a group is something we made up. Perhaps there are patterns with objects that happen to work in seemingly quantitative ways, but those objects themselves are not math.
I'm not trying to be rude, but can I have some idea as to your background? It doesn't affect the validity of what you say or make you inferior in anyway, it's just that I think I might be assuming shared familiarity with some foundational ideas and misleading you as a result.
Edit: If you don't wish to share of find it to be off topic, I'd be happy to continue on as is.
I do think it is somewhat off topic but I do know calculus and studied some number theory, it literally took me two pages to explain why 1+2=3 because I had to proof every step and label what 1+2 and 3 were and prove that those labels were accurate. In short I think math is logical, but I don't think it itself is the universe. I think they are the patterns we find in it recorded as symbols on a piece of paper.
Even if you look at the qnautinization of orbitals, you can still explain it qualitatively this way: "The probability of an electron reaches a point below possible measurability in certain areas".
What on earth sort of course would force you to "prove those labels were accurate?"
Did your number theory course include proof-calculus based classes (this is very different from your "calculus" classes and if you're not sure the answer is no) andshould I assume then that you don't have much experience with foundations of mathematics or ontology? Topology?
Edit: again, it's not problematic if you haven't. It just means that I'll be more careful about such topics.
If I label 1 as a', all its principals have to hold through. a' has to be a' and the properties of a' being a' cannot change.
In calculus you do prove things, but not like in number theory, you do simpler proofs, such as to test if a formula works, like if a formula you wrote is the correct formula for a set of numbers, it has be equal the same answer as what your deriving it from at an input of 1 AND x+1, or always the next number. I think there was some guy who wrote a rule about prime numbers, and he tested it with 1000 prime numbers, but he never tested it with "x+1", and so eventually someone found a prime number that the formula didn't work for. Or you do things like you prove trigonometric functions like proving tan is equal to sin/cos using certain properties and finding derivatives and etc.
No, I know what I'm trying to say, but perhaps I'm not saying it right.
A piece of paper is not math, even if you can put math on it, a pencil is not math just because you can right math with it. An atom is not math just because it happens to have patterns that you can happen recognize with numbers that we can write with a pencil on a piece of paper.
I understand what relabeling is, what I'm asking is what sort of course would require you to defend the labeling of 1, 2, 3 and the basic operators of arithmetic.
Proof calculus is not what you did. What you did was differential calculus with proofs. The phrase "____ calculus" is widely used in mathematics to apply to a large variety of fields.
The issue is not what you are trying to say. The issue is, pardon the assumptions, that you are trying to portray a much broader background than you actually have. The difficulty is that if, assuming good faith, you are responded to with the expectation that you have such a background, problems arise when the people you are speaking to respond with the assumption you know basic principles you have not yet learned.
Number theory would. Perhaps I'm not explaining it right, but a' is a' because of a specific property called the reflexive property which sometimes you have to state. I suppose I did hate number theory a lot and I only passed with a C, but it's hard for me to think I put all that work into it just to be wrong about something like this.
Perhaps it is, but just just goes to show number theory and "calculus with proofs" are different.
Ok, so aside from what I'm saying not actually being wrong, we're trying to discuss if every object in the universe is math, how does even having a background in number theory help that when number theory can't prove or disprove that? It's philosophy. Number theory is just the logic to prove things in mathematics. It seems like your trying to discredit my knowledge math so that you can say "He doesn't know math, so his philosophy doesn't work", even though math and philosophy are two completely different unrelated topics. I think at this point it is going off topic.
Can you prove that 3 x .333... = .999... ?
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I attempted to prove it, but then I realized it would take pages even in a simplified way, because I would have to prove not only that .333... is equal to 1/3 and that .999 is equal to 1, but also that .333.... and .999... are cyclic numbers.
I suppose if I said
x was equal to .333 then multiplied each line by 10, so
x=.333...
10x=3.333...
then subtracted the first line from the second line to make 9x=3,
10x=3.333...
x=0.333...
-
__________
9x=3
then divided x and 3 by 9 to make x=3/9 I could say 3/9=1/3 and then 1/3=x=.333 and 1/3 = .333 via the transitive property and use that to say that 3 x .333... = 3 x 1/3 = 1 = .999.... and I could probably get away with transitive property again on this forum.
Actually proving it with number theory though would take a lot of work, something I'm not willing to do for some forum post and neither is Macius probably.
I don't think you know what number theory is. It would not include proofs of real number equality. I have stopped responding to your posts because it has gotten exhausting, but please stop posturing.
Also, your proof is invalid.
Um, yes it can, but your always given some axioms but not always the same ones, and sometimes those axioms are "number ABC is a real number" or "XYZ is divisible by three" which you can use to say "If the real number XYZ is divisible by three, every permutation of XYZ is divisible by three." But let's say your proving a geometric theorem where two shapes share the same side, at one point you can say A of triangle BAC = A of triangle DAF; reflexive property" which is where you'd state A=A.
That's because it's not a real proof, it's more algebra than anything, something you'd know if you were actually reading my posts.
I'm not necessarily siding with paths... but you drop gads of jargon in a way that makes it sound like you're hinting you have some deep understanding of the math.