Flaws in modern day mathematics and their reasons/implications

[Sophius]

This is somthing I believe is ironic. That mathematics, a field that requires precise calculations with no room for error, should be flawed. What atre these flaws? I will give the two main ones.

First of all, we come to most heatedly debated point, the question of how can 1 contain infinity. Currently mathematics teach that hathere is an infinite amount of number between 0 and 1. This conversely means that the samme can be said or 1 and 2, however because 2 is 1+1, that means 2 is double infinity, which is not possible. Some mathematicians state that 0.99, 0.999, 0.9999, and so on is actually equivalent to one. However if we consider that those numbers can be easily represented in a reality based incrimential value not equivalent to one, that theory can be easily diregarded. Currently, the only way to have an explanation to this is to look at is from a philosophical standpoint, which is said not to overlap with mathematics(though ancient greek and roman philosophers would tend to disagree.) The reason why thta is however, is a completely different conversation altogether.

My second point, which is not a commonly debated one, is the fact that given probabilities are correct only in theory. Here is an example: If we were to posses a six sided die that is flawless in its design, we would say that it has 1/6 probability to come up with a certain answer. However to possess such an exact probability in real life is all but impossible. Let us say that there is a party going on in a penthouse apartment. Naturally, two elements in this party would be drinking and gambling. If a gambler were to roll a six sided die, we would say that their is 1/6 chance for each number. However, this theoretical probability does not include the possibility that a wandering drunk could bump into the table amd thus knock the die, which has not yet come with a result,out the open window. The die could then fall to the ground, land on its point, and shatter. A mathematician would say that such an occurence has such a low probability of happening(this is also debatable) that it can be discredited. However, if one were to factor in all possible occurences, the culimation would be signifigant. And, to expound on that, there is also theoretically an infinite number of possible outcome at any given time, thus turning the question of probability into a paradox.

What is everyones opinion on this?

[Balcerzak]

What is everyones opinion on this?

My opinion on this is that you're woefully ignorant of the concepts you're trying to claim are flawed.

You clearly have no idea of the proper mathematical concept of infinity, and seem to be unaware that there are indeed different classifications of 'infinity'. Your discussion completely leaves out any distinction between the countably infinite, the uncountably infinite, or how the cardinality of infinite sets is even properly handled. Throwing around terms such as 'double infinity' and misunderstanding even the basic concepts of how infinity works means that wasting much more time educating you on the matter is probably futile.

Your comments on probability are even more ridiculous. In cases where actually modeling the effect of many different low probability effects is important, there are methods of constructing appropriate probability distribution functions to handle them. Probability works well enough that while yes, your coin may be unfair, or your die may be loaded, but given sufficient trials, you can quantify these effects, decide whether or not they are relevant for the situation at hand, and deal with it.

Furthermore, your claim that mathematics actually needs to be 'exact' is suspect. While, yes, certainly it does in the realm of pure and abstract mathematics, and for constructing proofs the answer is 'yes', as soon as you get to the realm of applied mathematics or physics, and actually enter the real world, you use things like the binomial approximation, or truncating Taylor expansions, or (insert other method here) all the time and guess what, it works!

[Silvercrow]

Of course there are an infinite amount of numbers between 0 and 1, and just because 2 is 1+1 doesn't mean there are twice as many infinite numbers. Infinity is not actually a number, it's just a way of expressing that there are literally endless possibilities of number combinations. There are just as many endless combinations between 0 and 1 as between 0 and 2, or 0 and 50685068506. 0.99999~ can never actually equal 1 if you count it, but the limit as you continue to add 9s approaches 1 and so if you were to theoretically state that there was a number with an infinite number of repeating 9s, it would equal 1.

Obviously the statistical significance of random occurrences like the one you described are completely situational. But, in the situation you described, it has no relevance to the 1/6 probability. It doesn't matter if a die is rolled and the DEATH STAR mistargets and hits it with its laser beam and it explodes before the number can be read, there is still a 1/6 chance of rolling any number on the die, assuming it is weighted equally a (unlikely, but that deviation is not normally significant enough to chance the outcome of, say, a game). If the die falls onto the street and shatters, there is still a 1/6 chance that, had it not shattered, it would have landed on a 6.

There are flaws in mathematics, but neither of those two are examples of them. Those are both addressed and thoroughly explained in even basic high school algebra 2/statistics classes.

Edited August 19, 2010 by Silvercrow

[Soren37]

On your probability point, anything is possible. It just depends on how many tests you are trying to run. Yes, the amount of options you have may seem infinite, but that's why probability is such a broad term.

This video more accurately describes the phenomenon of coincidence, and how probability plays a big part, not supernatural guesswork.

On your first point, your theory does bring up an interesting point. I will agree about there being an infinite number of numbers between 0 and 1, and that the amount of number between 0 and 2 is twice the amount of 0 to 1. However, it depends on the numbers you are using. If I were to say that there were an infinite number of numbers between 0 and 1, you would agree with me, as decimals can go on forever. If I were to say that there was infinite amount of numbers between 0 and 100, I am not technically wrong, as there are still an infinite amount of numbers between the two. If I were to say that there were 100*infinite numbers between 0 and 100, then you know I'm shitting you, as we can only have one infinite, as infinite is a unlimited amount of numbers.

[Balcerzak]

... we can only have one infinite, as infinite is a unlimited amount of numbers.

Wrong. The set of real numbers between 0 and 1 contains more members than the set of integers.

[volkethereaper]

Infinity, as has been stated before, is not a number. It is a concept, an idea, and an impossible one to really wrap our heads around. As to probability, much the same can be said. The "flaws" of modern mathematics are simply the human inabillity to truly comprehend the abstractions of math. We will always have a difficult time doing this, simply because we can percieve that things like infinity exist theoretically, but we can't really percieve their true meaning.

[Sophius]

You clearly have no idea of the proper mathematical concept of infinity, and seem to be unaware that there are indeed different classifications of 'infinity'. Your discussion completely leaves out any distinction between the countably infinite, the uncountably infinite, or how the cardinality of infinite sets is even properly handled. Throwing around terms such as 'double infinity' and misunderstanding even the basic concepts of how infinity works means that wasting much more time educating you on the matter is probably futile.

Your comments on probability are even more ridiculous. In cases where actually modeling the effect of many different low probability effects is important, there are methods of constructing appropriate probability distribution functions to handle them. Probability works well enough that while yes, your coin may be unfair, or your die may be loaded, but given sufficient trials, you can quantify these effects, decide whether or not they are relevant for the situation at hand, and deal with it.

Furthermore, your claim that mathematics actually needs to be 'exact' is suspect. While, yes, certainly it does in the realm of pure and abstract mathematics, and for constructing proofs the answer is 'yes', as soon as you get to the realm of applied mathematics or physics, and actually enter the real world, you use things like the binomial approximation, or truncating Taylor expansions, or (insert other method here) all the time and guess what, it works!

It would be nice if my teachers in school were to provide these points. The only thing they ever gave me was avoidance. Thank you.

[volkethereaper]

Well, that's probably because your teachers are as uneducated as you in such fields. That is to be expected at the high-school level, but if you're in college I have but three words for you. Run. Run now.

[eclipse]

How much math have you completed?

(not being belligerent, just curious)

[Sophius]

I just graduated from the 8th grade last semester and I have completed the courses up through geometry.

Edited August 19, 2010 by Sophius

[Esau of Isaac]

How much math have you completed?

(not being belligerent, just curious)

He's fourteen, so he's probably just getting into Geometry.

Edit: I got the age wrong, but I was pretty close on the math. I know, I know, you are free to worship me.

Edited August 19, 2010 by Esau of Isaac

[eclipse]

Looks like you haven't hit the stuff where infinity is a boundary. If your school system is like mine, your next course will be Algebra II. I think that's where asymptotes are introduced (it's been over a decade since I took that subject). If not, you'll definitely see them in trigonometry. Asymptotes should be able to help you with the concept of infinity. I'm not going to attempt to explain this, because my memory's foggy.

Congratulations for doing better than 90% of my state's freshmen!

[Esau of Isaac]

Looks like you haven't hit the stuff where infinity is a boundary. If your school system is like mine, your next course will be Algebra II. I think that's where asymptotes are introduced (it's been over a decade since I took that subject). If not, you'll definitely see them in trigonometry. Asymptotes should be able to help you with the concept of infinity. I'm not going to attempt to explain this, because my memory's foggy.

Congratulations for doing better than 90% of my state's freshmen!

I think asymptotes were introduced in Algebra II, but they were glazed over and weak introductions. Pre-calc and (I think) later parts of trig are when one starts getting really introduced to them, to prepare for the total lack of awesomeness that is the middle portions of Calculus.

[Crystal Shards]

Asymptotes were introduced in my Algebra II class. Also, with regards of your possibility, it depends on if you're looking at an individual instance versus a group of instances. A drunk bumping into the table wouldn't affect that particular roll's chance of landing on a 6, but it would potentially have an affect on the outcome of a set of rolls. And really, depending on what you're doing, that would invalidate the set of rolls anyway, because there's an outside influence.

[RUSH]

Wrong. The set of real numbers between 0 and 1 contains more members than the set of integers.

Thats debatable.

One set includes an infinite amount of steps with a cap, one includes an infinite amount of steps (granted farther apart) with no cap.

Anyway

The one about the .99 continuing has always bothered me. It goes against everything math is about, such as impracticality in favor of exact correctness. Never is that number going to round above the decimal without rounding. I know about fraction proofs and whatnot, but I still disagree.

[Silvercrow]

Anyway

The one about the .99 continuing has always bothered me. It goes against everything math is about, such as impracticality in favor of exact correctness. Never is that number going to round above the decimal without rounding. I know about fraction proofs and whatnot, but I still disagree.

You don't need fractions to prove it, it's an asymptote just like was mentioned by a couple others. If you were to express the idea as a function, the limit of the function as x->infinity would be 1. So, if there were an infinite number of 9s, yes, it would actually equal 1, but that's not capable of happening in a way that's measurable. In the same way that, technically, if you faced a wall and stepped over half the distance, and continued doing so, you would never reach it, it's something that happens only theoretically, if you kept adding decimals of 9 after a 0 and expected it to magically become 1 it would never happen.

Edited August 19, 2010 by Silvercrow

[eclipse]

I think asymptotes were introduced in Algebra II, but they were glazed over and weak introductions. Pre-calc and (I think) later parts of trig are when one starts getting really introduced to them, to prepare for the total lack of awesomeness that is the middle portions of Calculus.

You mean the parts where it is not unusual to see limits from negative infinity to zero? Or the part where the trig functions come back?

Did you take a class called analytical geometry? That's where asymptotes became important, IIRC (I had a bad teacher for this class, so I don't remember much). Mine was offered in conjunction with trig.

[Balcerzak]

Thats debatable.

One set includes an infinite amount of steps with a cap, one includes an infinite amount of steps (granted farther apart) with no cap.

It's not even debatable. It's mathematically proven. While I don't recall the full rigorous proof, perhaps I can elaborate enough with what I do remember to convince you.

First, we must accept that the cardinality of all integers is the same as the cardinality of the counting numbers. This is easily done, as a one-to-one mapping can easily be constructed using a piecewise function

f(x) = {if x<0: (-2 * x) -1; else 2 * x}

The first half of the function maps all all the negative integers to the positive odd integers, while the second half maps all non-negative integers to the non-negative even integers.

Similarly, one can get from the natural numbers to the counting numbers just by f(x) = x + 1, to eliminate the zero.

Once we have that down, let us consider the following. We're given the interval from 0 to 1. Let us consider fractions with unit numerator.

Say, for all x elements of the counting numbers f(x) = 1/x. [NB: since zero is not present, this function is well defined everywhere].

All such fractions lie between 0 and 1 on the real number line. All of them. The function can actually be extended so that you can map all of the rational numbers in fact, not just those with unit numerator, but I forgot how at the moment, and don't feel like either googling or consulting my text. So now we can account for many other numbers in the unit line, like 3/4 and 7/9.

However... There is one set of numbers we cannot include, in any manner in our mapping. Irrational numbers, such as sqrt(2) or PI/4. The existence of irrational numbers is what makes (any nontrivial interval on) the real number line uncountably infinite.

[dondon151]

It would be nice if my teachers in school were to provide these points. The only thing they ever gave me was avoidance. Thank you.

Well, you aren't not even in high school yet, junior high school teachers do not tend to be very proficient in these subject areas, and it's pretty funny that your tone was one that challenges these concepts when you knew (hopefully) that you had no idea where you're talking about.

I mean, seriously. You're in 8th grade. There is only a very small handful of people at that age who might understand this to any significant extent.

[Fayt Zelpher]

Thats debatable.

One set includes an infinite amount of steps with a cap, one includes an infinite amount of steps (granted farther apart) with no cap.

Alright.

*Breaks out intro to Advanced Math textbook*

Proof that (0,1) contains more numbers (or, in more exact terms, that the cardinality is strictly greater) than {x: x is an element of the natural numbers}

Proof by contradiction:

Suppose that (0,1) has the same cardinality of N (the set of all natural numbers).

Then there exists a function F from N onto (0,1). (That is, that you can define a pairing between a single element of N with a single element of (0,1) such that every possible value in the range (0,1) has an associated value of N.)

Then, we can say that:

F(1) = 0.a₁₁a₁₂a₁₃a₁₄a₁₅...

F(2) = 0.a₂₁a₂₂a₂₃a₂₄a₂₅...

F(3) = 0.a₃₁a₃₂a₃₃a₃₄a₃₅...

And so on indefinitely, where a_jk is the k^th digit of F(j) (note that every choice here is arbitrary).

Now, suppose I create a new number, B, where B is defined as:

B = 0.b₁b₂b₃b₄b₅...

Where b_i =

7, if a_ii != 7

3, if a_ii = 7

Therefore, regardless of the choice of each a_jk, B will not be an element of the range of F (it will vary at the i^th decimal point). As a result, F is not onto (0,1), and the cardinality of N is strictly less than that of (0,1).

Yes, this is probably over most people's heads here, and it's probably not rigorous enough to be completely airtight, but this should be a good idea of why there are more elements in (0,1) than in {1, 2, 3, 4, ...}

(Note that because (0,1) contains the set {1/2, 1/3, 1/4, 1/5, ...}, it must be at least as large as {1, 2, 3, 4, 5...}. This proof shows why (0,1) has a cardinality strictly greater than N.)

EDIT:

I should point out that a simple mapping of (0,1) onto (0,2) using F(x) = 2x should demonstrate that the cardinality of (0,1) is exactly equal to that of (0,2). A strange but true property of infinite numbers.

Edited August 20, 2010 by Cocytus

[Crystal Shards]

Well, you aren't not even in high school yet, junior high school teachers do not tend to be very proficient in these subject areas, and it's pretty funny that your tone was one that challenges these concepts when you knew (hopefully) that you had no idea where you're talking about.

I mean, seriously. You're in 8th grade. There is only a very small handful of people at that age who might understand this to any significant extent.

Better for him to question and learn than question and shut himself up to learning.

[Kinata]

Funny, I've been doing a lot of thinking on infinite set theory recently. Anyways, the end of Cocytus' explanation confused me, so I'll give a clearer explanation. AKA, Cantor's Diagonalization:

If there was a bijection between (0,1) and the natural numbers, than we can write those real numbers in an infinitely long list (make sure there are no infinite strings of nines; all terminating decimals should have an infinite sequence of zeros):

.5345675...

.3293565...

.3979347...

.7787635...

.9767475...

.3678763...

.3457899...

And so on. Now look at the diagonal: .5277469...

Change every digit (once again, no infinite strings of nines): .6388570...

This number is in (0,1) but not on the list. Contradiction. QED.

And on the topic of probability, probability is not always correct, it is just a conclusion based on the current info. For example:

I have a bag with a red and a blue marble. I pick one. What is the probability that it is red? Currently, it's 50%.

Now I tell you that I will look down to make sure I pick a blue one. The probability is now 0%, yet the situation has not changed. The only change is that we have more info to get a better conclusion.

As for dice, I roll a die. It is loaded so one number is guaranteed. What is the probability I get 1? The answer is 1/6 because with the current info, each number is still equally likely.

[Fayt Zelpher]

Funny, I've been doing a lot of thinking on infinite set theory recently. Anyways, the end of Cocytus' explanation confused me, so I'll give a clearer explanation. AKA, Cantor's Diagonalization:

If there was a bijection between (0,1) and the natural numbers, than we can write those real numbers in an infinitely long list (make sure there are no infinite strings of nines; all terminating decimals should have an infinite sequence of zeros):

.5345675...

.3293565...

.3979347...

.7787635...

.9767475...

.3678763...

.3457899...

And so on. Now look at the diagonal: .5277469...

Change every digit (once again, no infinite strings of nines): .6388570...

This number is in (0,1) but not on the list. Contradiction. QED.

That's the same idea of what I was doing, except that the diagonal would have ended up being changed to

.7733777...

That is, when the digit obtained from the diagonal is not 7, it becomes 7, otherwise, that digit becomes a three.

[dondon151]

Better for him to question and learn than question and shut himself up to learning.

It's alright to question; it's an entirely different thing to report your lack of understanding as flaws in an ancient, well established field.

[Fayt Zelpher]

Some mathematicians state that 0.99, 0.999, 0.9999, and so on is actually equivalent to one.

Well, a second-semester course in elementary calculus is sufficient to prove this, actually.

Consider the number 0.99999999999999.... (infinite trail of 9's).

It could be said that this is the same as 0.9 + 0.09 + 0.009 + 0.0009 + ... (an infinite summation).

This summation is the same as the summation of x equals 1 to infinity of 0.9 * (.1)^s

It can be shown that the summation of a geometric series of the form a * b^x is equal to:

a / (1 - B)

Thus, the sum of that infinite series is equal to 0.9 / (1 - 0.1) = 0.9 / 0.9 = 1.