1. Points have zero dimension
2. A continua has an uncountably infinite number of points
3. All continua have the same structure and cardinality
4. Therefore it follows that all continua have the same length — Devans99

I think 4 doesn't follow.

Consider two lines AB = 2 cm and CD = 4 cm

Divide these lines i.e. find points on these lines by dividing them with k

1. k = 1; for AB, 2/1 = 2; for CD, 4/1 = 4
The point on AB is 2 and the point on CD is 4

2. k = 2; for AB, 2/2 = 1; for CD 4/2 = 2
The point on AB is 1 and the point on CD is 2

3. k =3; for AB, 2/3; for CD, 4/3
The point on AB is 2/3 and the point on CD is 4/3
.
.
.
n. k = n; for AB, 2/n; for CD, 4/n
The point on AB is 2/n and the point on CD is 4/n

As you will notice for every point on AB there will always be a unique point on CD i.e. the cardinality of the set of points on AB = cardinality of the set of points on CD. They're both infinite.

However, notice that a point on AB has a different numerical value to the corresponding point on CD. They are different quantities and so add up to different, not same, lengths.

As I keep telling you again and again and again, sometimes no matter how large your experiment is, it doesn't exhibit the pattern you mention. Sometimes you might throw the die 1 billion billion billion billion billion billion billion billion times and always get the same number, or never get some number. It is extremely rare, that's the only reason why you haven't noticed it. — leo

That unexpected events occur isn't an explanation for the issue I raised but...

This is wrong also, you throw the die in different ways that's why there is a multiplicity of outcomes, otherwise what you're saying would imply that if the die behaves deterministically it would always land on the same side no matter how we throw it, THAT would be the weird thing. — leo

I underlined the part that made sense to me. It squares with my explanation. Thanks.

@Andrew M@Harry Hindu

• Deterministic systems can behave probabilistically
• Ignorance or rather the impossibility of knowing was the actual impetus for the development of probability theory

Thanks all

Aristotle made a fuss about zero-dimensional points being the components of lines so I feel the question can be regarded as an open philosophical question. — Devans99

What were Aristotle's objections to points being zero-dimensional?

But this (questionable) maths leads to the conclusion that all continua have the same length - both a segments and its sub-segments are continua so they would both have the length 1. — Devans99

I don't get this part.
Are you saying that because there are an infinity of points in any given line that all lines have to be of the same length?

Is there something that still isn’t clear? — leo

I'm not saying a given outcome(s) is/are impossible. Perhaps I don't see the relevance of what you're saying to what is a actually bothering me. Kindly read below.

Randomness and chance would just be terms indicating one's ignorance of the relevant information for making correct predictions. — Andrew M

@Harry Hindu look at the part underlined.

I've given it some thought and I think you both are correct but not in the way you think.

Imagine a deterministic system A (a fair die with 6 sides). Once we have all the information on A we can make accurate predictions of how A will evolve. Deterministic systems will have specific outcomes right? There's nothing random in A and so however A evolves, everything in A will show a pattern and there won't be any variation in the pattern.

Please note that patterns are of two types which are:
1. Deterministic patterns. A good example would be gravity - there's a force and that force acts in a predictable manner.

2. Non-deterministic or probabilistic patterns. A die throw is effectively random but any sufficiently large experiment will demonstrate that the outcomes have a pattern viz. that three appears 1/6 of the time, an odd numbered face will appear 3/6 of the time.

Also bear in mind that a deterministic pattern will differ markedly from a non-deterministic/probabilistic pattern. The latter will exhibit multiplicity of outcomes will the former has only one determined outcome.

Imagine now that we lack information i.e. we're ignorant of factors that affect how A will evolve. We assumed A to be deterministic and given that our ignorance has no causal import as far as the system A is concerned, system A should have a deterministic pattern. However, what actually happens is system A now exhibits a non-deterministic/probabilistic pattern.

I will concede that there was a lack of information about system and that is ignorance but that has no causal import on A which should be exhibiting a deterministic pattern because system A is deterministic as we agreed. However, the actual reality when we do experiments we observe non-deterministic/probabilistic patterns.

pseudorandom — Andrew M

Correct. I wonder how one differentiates the true random from pesudorandom?

Very well put. — John Gill

I don't or rather can't practice what I preach :confused:

I suppose you can view a line segment as constituted of points or sub-segments. Whichever way though, the length of the constituents has to be non-zero. — Devans99

I'd like to continue the discussion if you don't mind because I see what you mean but I feel, given that mathematicians don't make a fuss about points being zero-dimensional, you're in error.

Points are locations. Location are distances. Distances are lengths. Lengths are spaces between two points. As you can see there's a definitional circularity: points defined in terms of lengths and lengths in terms of points and I think this is the reason why a point belongs to the list of undefined terms in geometry.

Anyway...

When we mark off a point, say C, on a line AB= 5cm, it needs an identifying label and that's its distance from one of the endpoints of the line. Suppose C is 3cm from A. We label point C = 3cm. C, the point, is zero-dimensional but remember it divides AB into two lines viz. AC = 3cm and BC = 2cm. AC + BC = 3cm + 2cm = 5cm. The line AB can be infinitely divided into infinitesimally small non-zero lengths and each length will always have a point associated with it. Can you now see that, since a line can be divided into an infinite number of non-zero lengths and each length has a point associated with it, there'll be an infinite number of points in any line. In some sense, a point is just a label/name for a length and while a length is a building block/structural component of a line the point is nothing more than the name for lengths.

Beyond finite instances, cardinalities are not numbers. They are equivalence classes. — John Gill

Thanks. I get it, more or less.

The basic theory of probability and statistics are pretty well established, but when it comes to practical applications there are some problems. Remember when vitamin E was popular for one's health? Then another study showed it had a negative impact on health. Vitamin C is great for this and that, then suddenly it wasn't.

The medical profession does not have a sterling reputation for statistical studies. Sometimes this is due to multiple experiments in which outliers are given undue consideration. Sometimes the experiments are poorly designed.

However, having said this I will tell you that probability theory still leaves me a little uneasy, even though its applications have been largely very successful. Probability waves in QM? Who'd have thunk? :brow: — John Gill

:smile: :up: If there's a hidden message in there I didn't see it

If you don’t look at the initial state, you may pick unwittingly the same initial state every time (or a member of the set of initial states that yield the same outcome) — leo

This is exactly what bothers me. It should be possible to bias the experiment towards a particular outcome. Yet this doesn't happen and the die behaves in a completely random fashion as is evidenced by the frequency of outcomes in an experiment of large enough number. Why?

However I don’t agree that there is a fundamental randomness that is introduced. — leo

:chin: The evidence for randomness is in the relative frequencies of outcomes in an experiment which perfectly or near-perfectly matches the theoretical probabilities.

It doesn't imply that. A number-generating algorithm simulating 1000 throws can be completely deterministic — Andrew M

. Yet all the outcomes taken together would follow a probability distribution — Andrew M

Because...

we can introduce randomness or more accurately pseudo-randomness into a deterministic system.

so if you always throw the die in exactly the same way you always get the same result. — leo

However if there are N different ways to throw the die (say 6 gazillion ways), and you throw the die once in each way, and the die is perfectly symmetrical, you will indeed get each side with frequency 1/6. — leo

The law of large numbers does not explain why if we don’t explain why that law works. — leo

It follows from the law of large numbers that the empirical probability of success in a series of Bernoulli trials will converge to the theoretical probability. — Wikipedia

We agree on the point that the outcome depends on and can be predicted by the initial state of the die. The rest of what you said depends on this and we see eye to eye on it.

If you'll allow me to keep things simple and not get into gazillions and combinatorics I think we'll agree on the following:

1. A fair die has 6 sides.

2. For the sake of simplicity assume that each side of the die {1, 2, 3, 4, 5, 6} is an event determined by six initial states {a, b, c, d, e, f} such that a causes outcome 1, b causes outcome 2, c causes outcome 3, d causes outcome 4, e causes outcome 5 and f causes outcome 6.

3. The [i[theoretical probability[/i] for each possible outcome when the die is thrown is 1/6

4. We have to accept that if we knew which initial state obtains we can accurately predict the outcome.

5. Now imagine you throw the die without looking at which initial state the die achieves. You will see the familiar result that each outcome is 1/6 of the total number of times the die is thrown. This concurs with increasing accuracy the greater the number of experiments that are performed.

6. 3 and 5 together imply that the die is behaving randomly

7. We also know that each initial state yields a accurately predictable outcome i.e. each outcome can be known given which initial state the die assumed

8. 6 says the die is random and 7 says the die is not random

9. Somewhere in the chain events, randomness was introduced into the system. The only place possible is at the time you put the die in one of the six initial states and this was random. This makes complete sense when you consider what you said:

Now let’s say there are 1 gazillion different initial conditions that yield the outcome three. That means you can throw the die 1 gazillion times in 1 gazillion different ways and always get the outcome three — leo

Your ignorance doesn't cause the dice to do anything. Your ignorance causes you to think of the world as probabilities and chances. — Harry Hindu

If it's as you claim, all in my head, how does the die know to come up three 1/6 of the time? Is the die sentient and after finding out I don't have the necessary information to predict, it does everything in its power to ensure that it behaves randomly in such a way as to match my probability predictions?

If it was all probability, then how is it not probable that you roll a 10 on a six-sided die? What constrains the possible outcomes? — Harry Hindu

It can easily be shown the P(10) = 0/6 = 0 = impossible. This has no bearing on why the die is behaving randomly.

There is such a thing as randomness and chance. They are ideas that stem from our ignorance. Like every other idea, they have causal power. It's just that you are projecting your ignorance/randomness/chance out onto the world where their only existence is in you head as ideas. — Harry Hindu

How does my ignorance cause the die to become random?

Separately, I must ask you this:

Are all random and chance events caused by our ignorance?

They are guesses and we guess because we are ignorant. — Harry Hindu

You seem to be saying that probability = ignorance but that would imply that there is no such thing as randomness or even chance.

If that's the case then consider:

1. A theoretical probability assumes randomness in its calculations. The theoretical probability for a three is 1/6

2. The die thrown 1 million times will show a three 1/6 of the 1 million throws


2 is exactly as predicted by 1 and 1 assumes randomness.

According to your claim then our ignorance led to the random behavior of the coin? How is this possible? How can my ignorance lead to randomness?

At that point the only apparent mystery that remains is why when you throw the die only 100 times, most of the time each side shows up about 1/6 of the time, and as I described in my previous post that can be explained with statistics, there is no need to invoke any fundamental randomness. — leo

but it is important to see that in some rare cases, even if you pick the initial conditions as randomly as you can, you can still get frequencies that are totally different from the theoretical probability (for instance getting the number three 1000 times in a row even though you have thrown the die in many different ways without knowing the outcome in advance, this is very rare but it can happen). — leo

I just read a very simplified version of the law of large numbers which asserts that as the number of probability experiments increases, the results of the experiment approaches the calculated theoretical probability.

So you're right that "unexpected" outcomes such as 20 threes in a row can occur in a 100 throws of the die. However, as the number of experiments are increased, say to a million throws, the frequency of threes in that million will be approx. 1/6.

At that point the only apparent mystery that remains is why when you throw the die only 100 times, most of the time each side shows up about 1/6 of the time, and as I described in my previous post that can be explained with statistics — leo

Did you mean the law of large numbers?

If yes then that implies the die is behaving randomly. Whence this randomness?

I wonder what political science has to comment about politics in science now that everything under the sun seems to be in the race to achieve scientific status or thereabouts.

The 13 scientists involved in the controversy are clearly a farsighted lot to realize the negative connotations of "supremacy". Even though the word in question is nowhere as offensive as "Hitler" we can see that their concern is quite legitimate. A stitch in time saves nine.

:ok: :100:

Yes, but randomly does not imply non-deterministically. — leo

This is what I want to discuss if you don't mind and thanks for your effort in trying to make me understand.

We have to revisit our assumptions:

1. Either a system is deterministic or it's random but not both

2. The die is a deterministic system in that with the necessary knowledge of the initial state of each throw we can predict every outcome accurately

3. Theoretical probability calculations has as a fundamental assumption that what is being calculated is random. The theoretical probability of the die showing three is 1/6

4. An experiment is done and the die is thrown 1000 times. In accordance with the theoretical probability we'll get three on the die approx. 166 times or 1/6 of 1000 throws

5. The fact that 3 and 4 agree with each other implies the assumption that the die is random is correct

Notice that 2 states the die is deterministic and 5 states the die is random/non-deterministic and this is a contradiction because of 1.

I'd like to give my own "solution" to the paradox:

A deterministic system can't be random and the die is behaving as if it is random. This implies that a random element was introduced into the system (the die) at some stage of the experiment (throwing the die 1000 times) and I think this happened when we chose the initial states of each of the 1000 die throws - all initial states were chosen randomly and so the outcomes conformed with the theoretical probability which makes the assumption that the system (the die) is random.

Do you agree with my "explanation"?

That there is a blurring of distinctions doesn't necessarily imply the absence of distinction. To think so would be to slide down the slippery slope from just a resemblance to exactly identical. Imagine if the courts did that.

That there's a dusk and a dawn doesn't mean night and day are same.

As for gender, essentialism and human nature, I think one needs to be aware of whether it's in terms of quality or quantity. Qualitatively there really is no point in saying what is or is not human nature. Nevertheless, statistically it's possible to say that the majority represents human nature.

Thanks for your patience. I don't think I've "discovered" anything. My question is more a reflection of my fundamental misconceptions on a subject, here probability.

Anyway you said:

1. The behavior of the die is caused by its symmetry and then you said

you throw it quite randomly, — leo

2. Just because the outcomes can be expressed as a percentage doesn't imply that the outcomes are probabilistic

Firstly, why did you say "you throw it quite randomly"? I would infer from it that it is necessary for randomness to enter into the system (the die) at some stage of an experiment.

Secondly it isn't the mere fact that I can express the outcomes as percentage but that these percentages agree with the theoretical probability which is possible if and only if the die is random. Yet, as you seem to agree the die outcome is deterministic in nature.

How do you reconcile the fact that the die is a deterministic system and yet behaves probabilistically? I'm as nonchalant about this as I would be if someone said s/he could predict the outcomes of random events.

There were years when I was convinced that our capacity for language must have evolved gradually as a complex form of communication, in accordance with the widely-held belief in evolutionary biology that organisms change slowly and incrementally via natural selection (with some exceptions, Steve Gould being an obvious example).

After reading Chomsky, I now lean much more towards the idea that not only did language not evolve gradually as a form of communication, but that language isn't communication at all.

I'm interested to hear if others, who have specialized in the evolution of language or are well versed in its literature, have considered Chomsky's ideas on this matter. I haven't seen much in this forum so far, although I am new to it. — Xtrix

One thing is for sure, we, all "higher" animals in fact, have specialized and dedicated organs for sound, our ears and that too as a pair. To the contrary, vocalization is always associated with the one mouth animals possess which has the primary function of eating. Our vocalization apparatus is just a secondary ability acquired through modifications of organs such as the trachea and esophagus as if it was a part-time job and not like a steady job one expects it to be if it was a priority.

That means hearing is more important and that can only be because there is vital information travelling in sound waves - prey/predator/water/shelter/etc. Doesn't this show that sound-based communication was critical to survival?

if so then language would initially have to complement the function of the ear i.e. it would be communication-oriented to capture prey and escape predators. Thinking would come later, much much later I believe.

Really? Consider N={1,2,3,...} and S={1,9} No 1:1 correspondence. You are missing "equivalence"

"A" proper subset? Not "The", I think. — John Gill

You quoted me out of context. Thanks anyway.

Wittgenstein's argument:

1. Meanings are of two types:
a. Meaning as use
b. Meaning through mental imagery and pointing out

2. Meaning type b involves to a large extent meaning type a

3. Change a contradiction of form (p & ~p) to simply p, ignoring ~p.

4. 3 above is simply a play on words

5. Isn't this actually a case of wrong continuation? If given all the definitions and rules then (p & ~p) is impossible.

6. yes in that if 5 is true it would be "natural" to be dissatisfied with any course of action.

7. There is nothing "natural" in math

8. Therefore, it's just a matter of switching gears and agree to make (p & ~p) "natural"

9. Does 8 make contradictions true or is "true" being used differently?

10. "True" is being used differently

11. Forget about true and false since "p is true" = p and "p is false" = ~p. Only assert and don't declare truth values.

12. When is a math based on 11 similar to standard or accepted math?

13. When "~" is being used differently

14. There are ironic statements where the literal meaning has an opposite truth value to the real meaning.

15. We can mark ironic statements (14) and get for example "|-- (p & ~p)" to mean contradictions are refutable or false

The argument doesn't work for me because at 15 he rejects what he tries to introduce viz. the contradiction (p & ~p) into math, after all "refutable" implies that something is false.

Also, notice how Wittgenstein avoids the claim that contradictions are true in 9 by giving us the option of "true" being misused.

:up: :up: to all

:ok: :up:

Now the prevailing wisdom is that [1] holds - a line segment is composed of an infinite number of zero length points. I cannot make any sense out of this. How can anything zero length (dimensionless) be said to composed something with non-zero length? This is the view Aristotle held. — Devans99

The only "explanation" I can offer is:

Consider a line of length 1 unit extending from 0 to 1.It can be repeatedly halved by multiplying with 1/2
Each multiplication will yield a point and a length corresponding to that point. For example the first halving will give us 1/2 which is a point and dimensionless but don't forget the distance from 0 to 1/2 which is a 1 dimensional length. You're failing to consider the length that corresponds to each point in a line. So, although points are dimensionless, the distance between points have a dimension viz. length.

Considered another way there are an infinite number of points in any given line but the line is constituted of the distances between these points and not the points themselves.

Can you have a look at what I said below. It seems wrong and right.

n(set of natural numbers) - n(set of even numbers) = infinity - infinity = zero because both are, well, infinite. — TheMadFool

Consider two infinite sets A and B of equal cardinality i.e. n(A) = n(B) = infinity
Shouldn't n(A) - n(B) = 0?

Yet n(set of natural numbers, infinite) - n(set of even numbers, infinite) = n(set of odd numbers, infinite) which is infinite indicating that infinity - infinity = infinity

injective, then there may be some elements of BB that are not mapped to by any element of AA;
surjective, then there may be some elements of AA that map to the same element of BB; and
bijective, then there is a unique pairing between elements of AA and elements of B — quickly

Thanks. I'll just stick to the simple.

What do you mean by insight?

If it (insight) complies with the lexical meaning then I think it's a product of clear, deliberate and focused thought.

However, the word "insight" has connotations that the word "understand" lacks which I'm only guessing is the comparison. These, to name a few, are "getting to the heart of an issue", and "glimpse the true nature of a problem". These are what I personally associate with insight.

From my own experience I've come to the realization that every and any issue we come across has what can be classified as critical and incidental elements and the latter are very effective distractions, red herrings as it were, leading you away from the former and obscuring the truth. Insight would, in this context, be the ability to identify such elements, separate the wheat from the chaff and once that's done you can hope for a more productive engagement.

So, yes, insight seen as I described above is germane to philosophy which is clearly complex enough to have more than an adequate number of distractions to sidetrack you in your inquiry.

I don't know how related insight is to pattern recognition but the latter brings to the table a very powerful tool to philosophical investigations because of how the recognition of a common pattern between two entirely different problems will enable the solution to one applicable, at least in principle, to the other. Nobody wants to reinvent the wheel.

Sorry for the long delay in my response but I was waiting for an epiphany of sorts. My mind just drew a blank so I'm going to work at this problem from scratch if you don't mind.

1. There is, more or less, an agreement that a die throw is deterministic.

2. The outcome of die throw can be calculated probabilistically e.g. probability of getting a 3 is 1/6

3. Each outcome of a die-throw can be calculated deterministically i.e. given the initial state of the die we can accurately predict each outcome

4. The outcomes of a set of 1000 die-throws can be predicted probabilistically e.g. 3 will appear approx. 166 times

5. point 2 agrees with point 4. In other words the die-throw is behaving as if determinism is false for the die

The problem for me is 1 and 5 contradict each other.

Some (@Harry Hindu) have said that probability = ignorance but that would mean that there is no such thing as actual chance and what we perceive as chance is a manifestation of our ignorance.

However, if that's the case 2, and 4 should be false but they are true and indicate the die is behaving as if determinism is false.

:ok: Thanks.

Literally typed this into Google and no coherent answer?

Is the question ill-posed, or does it mean anything, as it seemingly does to me, at least? — Wallows

It appears to me that there are two types of contradictions:

1. The obvious one that goes (p & ~p) in which the two statements of the conjunction have opposite truth values: if one is true then the other has to be false. This type of contradiction is used in reductio ad absurdum proofs which is the application of the law of noncontradiction ~(p & ~p).

2 A contradiction is a statement that is always false. Type 1 contradiction seems to be subcategory of this type of contradiction because (p & ~p) is always false.

How type 2 contradiction differs from type 1 contradiction requires a discussion of inconsistency. A group of statements is inconsistent if and only if the truth table of the group doesn't have a single line where all the statements are true.

You'll notice that if all the statements of a group of inconsistent statements are put under a conjunction i.e. joined together with the & connective, we will get a contradiction. After all for a conjunction to be true all conjuncts have to be true and that isn't possible with a group of inconsistent statements.

Where type 2 contradictions differ from type 1 contradictions is that it's possible for all statements in a group of inconsistent statements to be false while in type 1 contradictions the component statements have to be of opposite truth value.

How do we "solve" a contradiction?

1. Type 1 contradictions require you to negate the assumption that led to the contradiction.

2. Type 2 contradiction literally means whatever belief system you're operating from is logically flawed because it's impossible for all the statements that constitute it to be true at the same time. For instance imagine a belief system X composed of statements A, B, and C. If A, B and C are inconsistent with each other the conjunction A & B & C always evaluates to false, is a contradiction of type 2, which means X on the whole is a false belief system. Naturally, we have to discard X.

This has nothing to do with curly braces or other arbitrary syntactic phenomena unless you are committed to some kind of hardline formalism about mathematics. — quickly

I read a little of the Stanford Encyclopedia of Philosophy article on the topic.

It seems that Aristotle thought of an actual infinity to be akin to the infinite divisibility of, for example, a finite length in that it is "complete" and potential infinity to be something like the non-terminating process of adding 1 to any number and getting the next greater number.

I'm not sure what he meant, but a bijection is both an injection and a surjection.

But to your question: a set is infinite if and only if it is equivalent to one of its proper subsets. And this counters your argument. You are making the mistake of thinking about an infinite set as finite.

The proof of this actually refutes your idea: See for example here, Proof 1

You see with a finite set it isn't so: a finite set can not be in one-to-one correspondence with one of its proper subsets. — ssu

Thanks for the reply. I understand that another definition for an infinite set is a set that is equivalent (bijection possible) to a proper subset of itself.

I think I get it now. The proper subset of the infinite set itself has to be infinite. The number of elements in a proper subset B of a finite set A is necessarily less than the number of elements in the set A. n(B) < n(A) so long as A and B are finite.

When we then consider infinite sets X and Y such that X is a proper subset of Y then n(X) = n(Y).

It seems that a difference in terms of members, so long as the proper subset of the other, the parent set, is itself infinite, doesn't translate into a numerical difference.

Imagine A = {s, d, f, h} and the proper subset B = {d, f}

The difference between the two A - B = {s, h} = C can be numerically represented as n(C) = 2

However with infinite proper subsets of infinite sets, the difference is zero.

n(set of natural numbers) - n(set of even numbers) = infinity - infinity = zero because both are, well, infinite. In other words the fact that the set of even numbers don't contain odd numbers and that the set of natural numbers is the union of odd and even numbers didn't translate into a numerical difference like it does with finite sets.

7. Therefore, IFF what one wants aligns with the overall good of the world, God will give one what they want whether or not they pray for it. — Teaisnice

What about people praying for world peace or a cure for cancer or the like. These are unequivocally aligned with the overall good of the world and yet remain unrealized even to this day despite the earnest prayers of millions.

If anything this brings to light how, everything considered, improbable it is that god will answer prayers for personal benefit.

However, there's the issue of how distinct the individual is from the world. Isn't the world just made up of individuals? Doesn't the happiness of each individual, however that maybe achieved, constitute what you call the overall good? If yes, then god should answer our prayers whether it was for a brand new TV or world peace. If no, then that would mean the overall good is not the same as individual happiness which seems odd because the world to which the overall good applies to doesn't and can't experience the good or bad except in terms of the individual human beings that constitute it.

I guess I'm saying the overall good doesn't make sense if it's contrasted to the individual good in a way that tries to prove that only prayers for the overall good will be/should be answered to by god. It's like someone saying I'll help the Jedi but not Luke Skywalker or Yoda or Mace Windu. Sorry for the poor analogy but it gets the point across.

I find the concept of a dimensionless object difficult - it has no extents so it cannot have any existence - how can any sound reasoning performed with a non-existent object - assuming its existence (in order to reason with it) leads straight to a contradiction? — Devans99

I did a cursory reading of Euclid's definition of a point: "that which has no parts" which I suspect alludes to points being zero dimensional.

The usual way points are expressed in Cartesian plane is by an ordered pair (a, b) which to me means points are intersections of lines, in this case the lines x = a and y = b.

Lines are not dimensionless, they have length, so shouldn't be a problem for you. After all you seem to have an issue with the dimensionless and lines have a 1 dimension viz. length. However, lines don't have width i.e. lines have one and only one dimension which is their length. This won't be a problem for you either since even though lines lack a width, lines have a dimension, length.

Now, consider the intersection of two non-parallel lines. They intersect on their widths and not their lengths. Such intersections being points can you now see how points can be zero-dimensional?

No, Heisenberg's uncertainty principle says that the mass-energy of an electron that exists for a short time is uncertain. See Bruce Schumm, Deep Down Things, Chapter 4, under the heading The Living Vacuum.

Schumm's book is readable by the layman, yet goes into considerable detail. I have intentionally avoided discussion of these details because I know I don't have a good grip on them. What Schumm makes clear, and every author I have read makes clear, is that we know nothing about what goes on when particles interact with each other. We only measure the results, and only as statistics in the aggregate. — GeorgeTheThird

I was expecting Heisenberg's uncertainty principle but you forget that predictability is about aggregates of particles and not individual particles. The aggregate mass in the form of a a ping pong ball or the earth itself stays constant, allowing us to make accurate predictions about the motion of such objects.

However, it is classically true that A<BA<B iff there is an injection from AA to BB but no surjection. — quickly

Again no! — ssu

I think @fishfry said something to the effect that bijection has precedence of injection. Why?

I think @quickly understood my argument but s/he still has to explain @fishfry:

If there exists a bijection between two sets, even a single one, even if there are plenty of functions that aren't bijections, then we DEFINE the two sets as being cardinally equivalent. It's a definition, not a proof. — fishfry

The problem with your argument is that the described function is not bijective. Instead, you have constructed an injection from the set of even numbers to the set of natural numbers. From this, you are only permitted to conclude that the set of even numbers has at least as many elements as the set of natural numbers. If you combine this with an injection from the set of natural numbers to the set of even numbers, then you can conclude that a bijection exists. Because these sets are countable, this bijection is constructible (and is given by the standard map from the natural numbers to the even numbers). — quickly

Showing that you don't quite understand what others are talking about isn't a great argument. — ssu

Thanks for bearing with my stubbornness but have a look at what I say below:

A = {x, y, z} B = {r, s, t} and C = {l, m}

Cantor got it right if his claim is that quantity/number is simply an abstraction of what is common between sets - specifically the possibility of putting their members in a 1-to-1 correspondence in such a way that each element in one set is paired with one and only one member in the other set with no element in either set left unpaired. This is called bijection I believe.

[Assume the notation that n(A) means the cardinality or number of set A.]



n(A) = n(B) because of the bijection between them. One way for the bijection is as follows:

x ---- r, y ---- s and z ---- t

The argument that the set of natural numbers has the same cardinality as the set of even numbers, i.e. a bijection is possible, is exactly like the above with one exception - we're dealing with infinite sets.

So far so good.

Consider now the set A and set C. There is no bijection between them and the set A has one extra member that doesn't have a counterpart in the set C. We then say n(A) > n(C) i.e. set A is bigger than set c. The attempt to pair the members of sets A and C will look like:

x ---- l, y ---- m, z ---- ???

The set theoretic way of saying n(A) > n(C) is that set C can be put in a 1-to-1 correspondence with a proper subset of set A.

There's no issue with any of what I've said so far. To summarize we agree on the following:

Facts:
1. Two sets have the same cardinality if and only if there's a bijection between them

2. A set G has a cardinality greater than a set H if and only if the there's a bijection between set H and a proper subset of G

[Definition: a set K is a proper subset of a set L if and only if all members of set K are present in set L and the set L has at least one member which isn't a member of set K].



Since all of you are on Cantor's side it implies that you understand the argument that the cardinality of the set of natural numbers is equal to the cardinality of the set of even numbers. This result follows naturally from fact 1 above.

However, we still have to consider fact 2 by which we can determine inequality of cardinality of sets

The set of natural numbers N = {1, 2, 3, 4,...}

The set of even numbers E = {0, 2, 4, 6,...}

It's clear that N can be separated into two proper subsets viz. the set of even numbers V = {0, 2, 4, 6,...} and the set of odd numbers, D = {1, 3, 5, 7,...}

Notice how set E has a bijection with set V and set V is a proper subset of set N. If so, then in accordance with fact 2 above, n(E) < n(N) i.e. the set of even numbers is less than the set of natural numbers.

This presents a problem doesn't it? One of what I called facts leads to the infinity of natural numbers having the same cardinality as the infinity of even numbers while the other (fact 2) leads us to the conclusion that the set of natural numbers is greater, not equal, than the set of natural numbers.

Comments...

Sorry I should of said:

- A point has zero length according to maths definition
- But according to my intuition, a point must have non-zero length — Devans99

Do you suppose there's a reason why points are zero-dimensional?

How would we define distance? The beginning/end of one point to the beginning/end of another point? Why not just consider the beginning/end as zero-dimensional points?